My turn: Clearing the finish completely under a bridge leaves absolutely no margin for error if and when the bridge needs to be serviced in the future because it's lifting or splitting..... or the saddle slot has split with the saddle tipping forward - all things that can commonly happen, especially the bridge lift. I'll add that other common causes of bridges lifting are bridge plate damage between the pin holes from using cheap, slotted, pl*stic pins effectively creating a crease from hole to hole and distorting the top surface under the bridge so that it's no longer flat and the glue line gives. A dry guitar, something that we see in the hundreds starting right about this time of year with the woods expanding and contracting at dissimilar rates also stresses glue lines and can cause a bridge to lift. Another cause that comes to mind are radiused tops that have had a flat bridge "smashed" on top with lots of clamping can give too in time even if the manufacturer uses nylon pins not only to locate the bridge but as a hedge against bridge movement in time. Nylon pins are more commonly used in production than you might think as well. Even some of the top tier small builders use nylon pins.
As such bridges need to be serviced in time like it or not and some repair person like me may have to be inserting a pallet knife in-between the top and bridge on one of your creations. Flakey finishes, too much heat in the glue softening process causing finish to bubble, not reading top run-out properly are all common reasons why if there is NO margin for error with the "cosmetic boundaries" of how a bridge is glued on the repair may show OR require unnatural acts to repair/touch-up finish when none of this was necessary.
Ken welcome to the world of f*ctory guitars if you are noticing that f*ctories don't clear much finish under their bridges. Not only is this an opportunity for small builders to outperform f*ctories in yet another respect it does not serve even them well when they have to pay me 7 years out to reglue a bridge under warranty.
As such regardless of who likes to clear ALL the finish from the entire bridge footprint I don't like the practice either and it clearly does not fit in well with the concept of serviceability down the road.
Going back to clearing finish under the entire bridge. Ever see cold creep of a bridge exposing a glue line or bare wood line on the back edge and lifted, mounded/plowed..... finish on the front edge - we have..... Had the entire bridge foot print not been cleared the back edge would not be something that is glaringly obvious.
For those of you who may not do repair work once bridges are removed the wings often curl and in order to get the bridge flat again some bridge height is lost on the belt sander getting the bottom flat again. This can also reduce bridge height removing material from the bottom to get to a flat state including the curled wings. Some bridge designs such as mine...... where there are NO vertical sides, ends, etc to the bridge shape removing more bottom material because of curled wings or even a curled bridge (that happens too....) WILL reduce the outside dimensions of the bridge making it smaller. If the bridge foot print is entirely devoid of finish the bridge can no longer be reused and has to be replaced. On the other hand if some finish is left under the bridge this can be avoided.
Now let's talk about how much finish is left under the bridge but before we do I'll reiterate that I once calculated up to 40% of bridge gluing surface being left under a major manufacturer's bridges that is finish that was not cleared - not......good....... especially on the back edge.
With the rabbiting that we do the amount of finish left under the bridge is a ledge that may only represent around .005" inward from the bridge perimeter. Do the math and you will see that we are still using around 98% of that 100% of bridge foot print that is available if we remove ALL finish and pocket the bridge in a well devoid of finish.
Regarding how wings can be problematic to get down and keep down that's where Luthiers have to earn our pay. With our rabbiting prior to the rabbiting the bridge is fitted to the top and this means imparting a radius if one is present and necessary. We do this so often these days that for us it starts on a belt sander using the ends of the platen where it's curved to match the top radius. Next the bridge is trial fitted and if it fits well in terms of the radius it's off to the rabbiting jig.
After the finish is cleared to around .005" from the edges of the bridge perimeter (and yes this margin for error is so very small that I have trouble even seeing it...., Dave does not....) the bridge is trial fitted again. We want the wood to wood contact to be that 98%ish of cleared foot print AND we also want the upper lip of the bridge rabbit to be sitting down on the finish ledge with NO gaps. With observation and trial, dry clamping we observe the fit and then judiciously scrape with a single edged razor blade the proud areas on the bridge bottom permitting anything sitting up where we don't want it to to sit back down and behave.
We don't rely on lots of clamping to make a bridge fit well. We want it to fit pretty well with NO clamping and then with clamping and firm clamping as well it all settles into place nicely.
Just like necks bridges need to be fitted, always!
By the way the guitar in the picture was picked up last Wednesday after being with us 2.5 days.... It belongs to an aerospace engineer and he bought it used so he could not speak to if the bridge had lifted once or more before. It is about 20 years old, he knew that much... and what likely caused this bridge to lift other than the terrible job.... of attaching the bridge either in a previous repair or from the f*ctory..... (I personally suspect previous repair mostly because of the use of "brown glue") was that this owner let the guitar dry out. The fret ends were proud, top caved, etc. all the signs of a dry guitar and he admitted same under hot lights and water boarding.....

Maybe we could use HHG boarding..... but the open time is too short....
Bridges need to be properly fitted and if you struggle with getting the wings down the thing is not yet fit properly and needs more attention. In my experience properly fitting a bridge on a severely distorted top can take more time than any other operation in the removal and reglue process.
Professional standards, at least our's do.... dictate NO gaps, none anywhere ever when regluing a bridge and how one gets to that goal takes lots of experience AND lots of opportunities for experience as well. The folks who seem to do the best work here are the folks who have over time gotten comfortable with their methods for better or worse but what works for them.
Repairs are very different from building too IME because when we build we build in our shapes and know what they are and where they are. With guitar repair the top can be a very different shape than when it was built making bridge fitting more difficult.
Although this may be an academic conversation for some... this is our life, every single day we may be either regluing a bridge or one is in the wings waiting for it's turn in on the bench. As such methods are developed over time and by time I mean a decade or more so that we can see what if anything may come back or not. We have had excellent success with the rabbiting method AND we are not the first or the last either to nix the clearing of ALL finish under a bridge and instead develop a hybrid approach attempting to exploit the best of all worlds with all considerations in mind. We remove around 98% of the finish AND address the issue of a flat bridge spanning a .003 or more finish ledge where our bridges because of the rabbit make wood to wood contact immediately where the finish is cleared. A flat bottom bridge will not for some distance inward spanning the finish ledge. The thicker the finish the more wood to wood contact area is lost.
Collings guitar who IMO likely produces some of the finest f*ctory instruments on the planet uses the rabbit technique too. Even on the OLF this method is used with Terry Kennedy and several others. I used it as well once I learned it but prior to that I used the clear the entire bridge patch method. By the way clearing the entire bridge patch with black topped guitars....

was not a great choice. SO far so good but come bridge reglue time something is going to show if not touched-up.....
CNC has absolutely nothing to do with anything that we are doing or describing here. Instead we are speaking of a method that works very well even over time, is used and embraced by other pros, AND may not be for everyone because you also need to be highly skilled in fitting a bridge even to the point of using a scraper to correct a .001" variance.
Oh yeah.... let's also address going forward or that all important consideration of serviceability in time. The repair for this guitar is NOT a good example of anything when it comes to the ease of future repairs but it most certainly is serviceable going forward anyway.
Typically as Dave indicated we don't route finish off obviously because you can see what happened either by a previous repair person's hand or the f*ctory here in this example. We use very sharp chisels and are only removing old glue exposing the wood never wanting to remove any wood.
I triage hundreds of guitars annually and it's been my experience that more often than not when a client guitar has a lifting bridge they are not even aware of it unless it's an extreme case which happens too. So we point out that the bridge is lifting, grab a .0015" feeler gauge and show the client that it can be inserted under the bridge.
When a bridge lifts the pallet knives are inserted being mindful of runout and other considerations that may be specific to that instrument. With the rabbiting of the bridge bottom the bridge has already lifted...... making palet knife access simple and easy. Bridges are not typically removed because they are not lifting BTW although that can happen too if a bridge needs to be replaced because it's splitting or the owner wants a replacement perhaps to replace a bridge that was already replaced in the past with non-original woods. I recall a 30's Martin that had originally had a rosewood bridge, received an ebony bridge in the 60's and the owner wanted it returned to a rosewood bridge.
Anyway in terms of getting a pallet knife in there and the serviceability of the repair when this bridge will need servicing in the future it will likely be because it's lifting again 40 years out when all of us are worm food. Because it's lifting palet knife access is easy and no harm was done in terms of serviceability. Reading runout will be key too.
Alternatively clients may not wish to pay $500 plus for a top to be resurfaced, finish touched up, and the bridge glued on as it should have been in the past without a .035" deep hole in the top. Instead this client paid less than half this amount and also received a complete set-up, new strings, and was and is happy as a clam. And that boys and girls is pretty important to us, always!