The flaw of this is that when two opponents have both used all of their missiles, they're no longer an option to break the stalemate. Instead, it can end up being rather unfun as the two ships slowly plink away at each other until one eventually runs out of CR or takes enough damage to die. Alternative mechanics like Burn Drive or having fleet-scale combat alleviate this, but the problem eventually comes around on the macro scale as well - when both fleets have used up all of their missiles, then it's a different kind of battle.
You cannot solve this by adding regeneration to missiles.
Have you tried it? Not to keep harping on it, but I have - and regenerating missiles do exactly the opposite of keeping a stalemate going. It means that if either side or both sides fail to break a stalemate with missiles, they can try again later instead of being locked in the no-missiles-on-either-side stalemate you describe.
Also the idea that missiles are simply "push button, ship dies" is ridiculous. The windows in which missiles are guaranteed kills worthy of their OP cost and limited ammo are extremely minute and get even more so as more ships get added to the fight. It also takes significant boosts from skills to make that possible reliably (even within those small windows) and that's a skill issue.
Also, regenerating missiles doesn't eliminate that (player) skill required - just because your missiles will eventually come back doesn't mean a bad launch by one side doesn't represent as significant disadvantage. It's still a huge chunk of threat gone for a significant amount of time.
Look at this way: Jousting for missile position is a significant combat high. It's skillful, it's impacted by everything from ship design to in-combat maneuvering, which ships are escorting which others, it requires timing - it's one of the most intricate parts of combat. And it happens once. Possibly doing nothing. And then none of this happens again. Your PD choices, that careful maneuvering - it's gone. The whole rest of the combat you are fighting with ships that are only half as interesting. Regeneration (and of course that requires a complete rebalance of missiles - they need it anyway) means one of the best parts of combat can happen multiple times and you get to fight with the whole ship longer instead of only until you launch your missiles.
The fact that one of the best ways to use missiles is to never use them because that hurts the AI more than empty racks is to me a clear sign that something is very, very wrong with missiles.
This is a really good analogy of how missiles can be a burst damage skill-based mechanic. They are balanced this way in my mod and are very satisfying to use. Regeneration is actually a crucial aspect to this balance both in terms of how the A.I uses them (muuch more skillfully when not conserving) and how limited or powerful the player can feel when using them.
In my mod, missiles usually come in clips of 4-6. They are one of the only weapons that generate flux (siege or strike-type ballistic and energy weapons do too) and they have a fast firing rate. They regenerate far more slowly, so burst dps is around 1500 but sustained is around 30. It takes roughly 30 seconds in combat to reload an entire clip, but you can trickle out individual missiles too to exert pressure on enemy defenses as needed (The ai usually does this after an initial blitz style strike to attempt to overwhelm you and then keep pressure on you by trickling out afterward for example).
So missiles become the ace-in-the-hole weapon that really dictates the flow of combat alongside similar-styled strike weapons. This makes missiles powerful almost universally in combat and ships of course are usually limited on missile slots for this reason. They are far more valuable, as is their related combat skills. This is balanced out by the individual ship's flux cap and dissipation (for reference, two small launchers firing all missiles in their respective clips maxes out the typical combat frigate's flux bar) that sometimes makes a less damaging, but constant and non flux-generating assault weapon more attractive to a ship load-out (an assault weapon typically deals around 75dps, so no burst but flux-free and double the sustained dps of a missile weapon).
Point defense, in turn, becomes crucial to defending ships from being focused on and picked off. A ship without pd or at least an interceptor escort (they are much smaller, faster and agile in my mod and great for a pd screen) is pretty easily overwhelmed and the ai is actually pretty efficient at choosing appropriate targets.
Honestly I was pretty skeptical that it would prove anything other than one-dimensional and quickly boring when I first played around with it, but as it turns out, the excitement from "missile jousting" (great analogy!) is pretty great and I've had a lot of fun testing it so far! This is, of course, a biased opinion as I'm developing this mod around my own personal playstyle and a more "star wars" type military doctrine in combat. But, I figured I'd share my findings since the question of actual working models was posed and I happen to have one
For anyone curious and to answer the question ahead of time, my mod is currently unavailable for public download and scheduled for a working download shortly-ish after the fighter changes are released in the next scheduled update (no clue when that is). Those changes are absolutely required to get a proper fighter/ship ratio in fleet spawning without going to the trouble of making custom scripts for everything. I've gotten pretty close in tests, but too many things still fail to hold up and kind of ruin the experience I'm going for. Hopefully soon though! Can't even begin to explain how excited I am for the next update!