I've got an A1200 in a Power Tower, with monitor, 4 gig hard disk, 32 meg fast memory, 030 accelerator, proper Amiga keyboard, 24x CD-ROM drive and a CD-writer when I need it.
I've also got a couple of A500s, with Action Replays 2&3, some A600s and a CD32.
I used to degrade old games on the A1200 with something like Relokick for the floppy games and Embedder (or others) for games on hard disk. Most worked okay, except for ones like Shadowlands and Rainbow Islands. But then WHDLoad and JST came along and lots of these old games started to work fine from the hard disk.
So, RCK, you can run nearly all games on an A1200.