There is one other check that should be made concerning Model M pistols, particularly older ones or ones (like that in the photo) that show some evidence of corrosion.
The Model M used a two piece firing pin (because a one piece pin would be too long to feed into the firing pin tunnel). A fairly strong firing pin spring is inserted in front of the "front" firing pin. I have seen more than one Model M where this firing pin spring had become fossilized. As a result, if one dry fired the pistol (the only way to store it with the hammer down), the front firing pin becomes stuck in the firing pin hole in the breach face -- and effectively you have converted the pistol to an "open bolt" mechanism that will AD the next time someone drops the slide over a charged magazine.
The test: dry fire the pistol, then lock it open and examine to see if the firing pin is protruding through the firing pin hole in the breach face.
The fix: At least as of a couple of years ago, replacement firing pins and firing pin springs could be found from the usual sources. In any event, a replacement spring (which is usually all the needs to be replaced) can be fashioned from a spring designed for something else. Replacement requires driving out the firing pin retaining pin (after field stripping); the pin drives out from left to right. (Be sure to use a cupped drift pin, so as not to mar the slide.) Tap the back of the slide on a bench block and the two parts of the pin and the spring should come out.
Examine the front pin for burrs, and polish if needed but do not remove any metal. Cleaning the firing pin tunnel is a bit fiddly, as you can't just run a straight implement into the tunnel, but I have done this with a .22 patch and a bent paper clip. After cleaning, the tunnel should be left dry.
Insert the new spring, front firing pin, and rear firing pin. With a little finesse, you can push the rear firing pin far enough forward to get the firing pin retaining pin started (inserts from right to left). Start the convex end of this pin from the right so that it will show on the left.
Now re-do the dry fire test. After dry firing, no part of the firing pin should be visible through the breech face.