X5000 etc..., be stuck in a regular ATX tower and be loaded with Amiga OS 4 or something to play games natively?
No and yes, all of the games you listed won't run on AmigaONE's (naively), AmigaONE's are almost like PC hardware (with exception of PowerPC cpu part), but is running a special PowerPC version of AmigaOS. But most of old games don't need the OS, it needs the Amiga hardware.
There are only few games that depends on the OS, and not on the hardware, and this games are mostly the games that where not originally Amiga games to begin with.
For most old games, emulation is required, simply because Amiga there where no Audio drivers back then, back then games where accessed the hardware directly. audio drivers came late to AmigaOS, some time in 1992 i think, but even then it was only really for audio expansion cards. the same with graphics.
Also can this theoretical setup use TCP/IP and access modern Internet routers via wi-fi etc...? And do basic USB peripherals work like a wired mouse etc...? I'm after that native experience but using modern kit for a modern Amiga flavour. Can this be done?
This is what AmigaONE's / AmigaOS4.1 are for, a modern Amiga experience on modern hardware, but with touch of the compatibility, if you did not use workbench on your Amiga500, then this is a completely new experience for you.
If you just like to play games, I think MIST is better option; MIST is more like your Amiga 500.
AmigaONE's are more like Amiga2000/3000/4000, desktop/tower Amiga machines.
