PlayStation 2 emulation is nowhere near accurate. Using the top "king" PCSX2 they're not going to be incorporating a method of accuracy due to too much time. Source: https://github.com/PCSX2/pcsx2/pull/3181#issuecomment-568209562 DobieStation is next best bet... when it gets up and running and playing games at full speed as they're not going for hacky solutions.
PSP emulation with PPSSPP is better with visual quality, however with the methods incorporated loads are horrendously wrong to an actual PSP (besides the other issues with framerate being also wrong).
Xbox360 with Xenia. It exists, nothing more to write.
Wii with Dolphin is the best contender for closest to actual hardware, unfortunately it's beating an actual Wii (and vWii) by not dropping frames and having that little edge in loads. Dolphin is still best case in getting more accurate to a Wii.
Since you didn't specify exactly which one you were after I broke it down for you instead.
It's not a question whether you encounter any problems with the game but whether the emulator is running the game almost exactly like a real console would. It's not necessarily something you'd notice in casual play but there are many aspects that play into sufficiently accurate emulation to have speedruns be comparable to consoles.
Emulator runs are now allowed on the leaderboards.
You can use Dolphin 5.0 to play the Wii version. Dolphin loads faster than any legitimate platform, therefore emulator runs will remain hidden by default but can still be submitted without any issues.
Further, you may submit runs emulating the
