running games from internal storage is pretty reliable on the PS2, yeah, I’ve got an IDE to SD card adapter in mine and it works great
PS1 games are a pain, though, yeah, there’s no way to put the PS2 in true PS1 mode while still reading from the internal drive, so your only option for that is emulation. the emulation (which, to be clear, is based on a hack of a Sony emulator, and said hack is entirely unmaintained since ~2017) is okay if your game is compatible, but it’s much less common than with PS2 games