It doesn't make sense that a werewolf could be killed by ordinary bullets

For me, this sort of thing doesn't ruin my enjoyment of a movie; it's just something that's fun to discuss. AAWIL is one of my favorite horror movies regardless.

The full moon would be the continuously running background process, i.e., it's what triggers the transformation to the werewolf form in the first place. Tissue damage would be a deviation from that form, and as long as the full moon is still out, it should trigger the transformation back to its full (undamaged) werewolf form.

Except, it's not an ordinary animal, it's one that was supernaturally created in a matter of about two minutes. Tissue damage from a bullet would heal almost instantly at that rate of generation. No organism in real life can generate tissue at anywhere near that rate because it would require an enormous amount of energy, probably on the order of megatons of TNT equivalence.

As a side note, David wasn't shot in the head or the heart. He only had two bullet wounds, both of which were in the torso, and neither of which were inline with the heart:

https://i.imgur.com/roIHqaF.jpg

And when he was shot he was in an alley facing the police officers, and he was on all fours and lunging forward, so there's no way that either one of those bullets would been angled toward the heart. Both bullets would have angled toward his waist, i.e., if they exited, the exit wounds would have been in the area of his lower back, or possibly even lower.

He was shot and killed by cops in an urban area who were responding to what they thought was a vicious dog or other dangerous animal loose in public. Why would they have silver bullets? The movie indicates that the city people didn't know about / believe in werewolves to begin with, unlike the country people at the tavern who kept their knowledge of werewolves a secret from outsiders.

Also, silver bullets aren't something you can just go and buy at a gun shop (silver is very expensive and not a good bullet material, therefore there's no general market for them). They have to be custom made, and finding someone to make them probably wouldn't be easy, because people who cast bullets cast them from lead / lead alloys and their melting pots don't get anywhere near hot enough to melt silver, and people who cast silver probably don't have bullet molds, because it's normally done for jewelry and such, not bullets. You could have a machinist make them on a lathe from silver rods, if you wanted to spend the money, not only for the silver (much of which would be wasted, i.e., end up as metal shavings on the floor) but for the labor; it would be a lot more time consuming than casting. Then you'd need to get them loaded into cases, so you'd need to find someone else to do that if you didn't have the equipment/supplies/know-how to do it yourself.

"Why only a silver bullet? why not any bullet?"

I explained in my OP why an ordinary bullet being able to kill a werewolf doesn't make sense. As for why a silver bullet, it doesn't need to be a silver bullet; the writer could make up whatever he wants as long as it makes sense within the context of the story. Silver is an obvious choice because it's a longstanding part of werewolf legends, and since it kills due to some sort of supernatural effect, it makes sense that it could kill a supernatural creature. If you're going to make up something other than silver, in order for it to make sense it has to either have supernatural effects or be able to do far more damage than an ordinary bullet, such as high explosives.

"Other werewolf films like Wolf and Bad Moon it didn't take a silver bullet to kill a werewolf."

It didn't make sense in Bad Moon either (even less sense than in AWIL, because the Bad Moon werewolf was even bigger), though it's not much of an issue in Wolf, because those were just "wolfmen" in that movie, i.e., their transformations didn't involve adding much mass to their bodies.

Why silver bullets?
Silver bullets would have more penetrating ability than lead as there is not as much flattening on impact. Looking at the transformation as the OP states, one would expect the werewolf to be tougher or more resistant to normal bullets.

It dates back a lot further than that:

c. 1640 the city of Greifswald, Germany was infested by werewolves. "A clever lad suggested that they gather all their silver buttons, goblets, belt buckles, and so forth, and melt them down into bullets for their muskets and pistols. ... this time they slaughtered the creatures and rid Greifswald of the lycanthropes."

Temme, J.D.H. Die Volkssagen von Pommern und Rugen. Translated by D.L. Ashliman. Berlin: In de Nicolaischen Buchhandlung, 1840.

"Why not?"

I've already explained "why not" in my OP. Did you only read the subject line?

"It's still an animal."

It's not an ordinary animal, obviously.

"It doesn't make sense that silver bullets would kill it, but not lead bullets."

Yes, it does, and I've already explained why. Again, did you only read the subject line?

"It doesn't make sense that a man can transform into a werewolf either"

That doesn't matter because that's the premise. Once you've established your premise, regardless of whether the premise is actually known to be possible or not, everything in the movie should makes sense within the context of that premise.