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.

This is PS2 emulation, where most engines were still bespoke and every hack in the book was still on the table.