As far as I know, static recompilation is thwarted by self modifying code (primarily JITs) and the ability to jump to arbitrary code locations at runtime.
The latter means that even in the absence of a JIT, you would need to achieve 100% code coverage (akin to unit testing or fuzzing) to perform static recompilation, otherwise you need to compile code at runtime at which point you're back to state of the art emulation with a JIT. The only real downside of JITs is the added latency similar to the lag induced by shader compilation, but this could be addressed by having a smart code cache instead. That code cache realistically only needs to store a trace of potential starting locations, then the JIT can compile the code before starting the game.
Yes, but in practice that isn't a problem. People do write self modifying code, and jump to random places today. However it is much less common today than in the past. IT is safe to say that most games are developed and run on the developers PC and then ported to the target system. If they know the target system they will make sure it works on the system from day one, but most developers are going to prefer to run their latest changes on their current system over sending it to the target system. If you really need to take advantage of the hardware you can't do this, but most games don't.
Many games are written in a high level language (like C...) which doesn't give you easy access to self modifying code. (even higher level languages like python do, but they are not compiled and so not part of this discussion). Likewise, jumping to arbitrary code is limited to function calls for most programmers.
Many games just run on a game engine, and the game engine is something we can port or rewrite to other systems and then enable running the game.
Be careful of the above: most games don't become popular. It is likely the "big ticket games" people are most interested in emulating had the development budget and need to take advantage of the hardware in the hard ways. That is the small minority of exceptions are the ones we care about the most.
How many PS2-era games used JIT? I would be surprised if there were many of them - most games for the console were released between 2000 and 2006. JIT was still considered a fairly advanced and uncommon technology at the time.
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JIT isn't _that_ common in games (although it is certainly present in some, even from the PS2 era), but self-modifying or even self-referencing executables were a quite common memory saving trick that lingered into the PS2 era - binaries that would swap different parts in and out of disk were quite common, and some developers kept using really old school space-saving tricks like reusing partial functions as code gadgets, although this was dying out by the PS2 era.
Emulation actually got easier after around the PS2 era because hardware got a little closer to commodity and console makers realized they would need to emulate their own consoles in the future and banned things like self-modifying code as policy (AFAIK, the PowerPC code segment on both PS3 and Xbox 360 is mapped read only; although I think SPE code could technically self-modify I'm not sure this was widespread)
The fundamental challenges in this style of recompilation are mostly offset jump tables and virtual dispatch / function pointer passing; this is usually handled with some kind of static analysis fixup pass to deal with jump tables and some kind of function boundary detection + symbol table to deal with virtual dispatch.