So as best I can tell magic is not allowed in Combat Biker and Urban Brawl, but cybernetics are. Would this include disallowing Adepts?

Adepts work by magic, so I'd say yes, they'd be disallowed.

And there'd be at least an aspected person capable of confirming, with a very high rating in assensing, if not a few spirits doing the job.

I know for a fact that magic is allowed at least in the European version of Urban Brawl. There's explicit mention of a magical heavy, Felix Winkler (a full magician), playing for the Viennese team.
Could be a disconnect between writers, or a difference in European rules. According to the Shadowhelix, it was the latter: magic, apart from adepts, was disallowed until the 63/64 season, including magical healing. Afterwards, these rules were relaxed to only forbid manipulation and illusion spells, as well as foci.
The writeup implies that physical adepts were allowed before that.

The writeup for Combat Biker in Shadowbeat doesn't mention magic at all, but seeing as most players are equiped with a vehicle control rig, and that it's forbidden to attack with anything other than with sanctioned weapons, I'd say that physical adepts are one of the few viable ways to bring magic into the game.
That is, of course, based on the original writeup, during which time a rig still cost humongous amounts of essence, and could have changed in the 70s, where it's a marginal investment.