OP, I can see where your mate is coming from; you break criminal law by doing something illegal, and you also break common law with going something illegal. However one must distinguish between the two forks of UK 'justice' – namely criminal law (i.e. defined in statute, carrying fixed punishments) and common law offences (i.e. historical crimes that haven't yet got round to becoming statute – so the more weird and wonderful crimes are still common law, but you can still be fined / imprisoned). I suppose in a few hundred years there will be no common laws.
So yes, cycling on Public Footpaths is in a sense 'illegal' but only under common law not criminal law, but its generally accepted that you only do something illegal if you break criminal law only because there are few common laws around to break
As for why cycling on footpaths is trespass? The simple answer is - because it just is; in a piece of circular (il)logic, because it's not criminal law you can't go to court to dispute it or redefine the statute. If it was criminal law it would be exceptionally difficult to prove damages / etc, so chances are it would NOT become criminal law and dropped.
You can see the difficulties above in trying to define 'trespass' and all the why's and wherefores, but it just shows it's in a sense an exercise in futility because unless it's a criminal matter, it just is.