Mmm... some "interesting" debate there but still none the wiser. When I'm running along this path, the dog will be at my heel so unlikely to stray outside the designated confines of the footpath. Walking is a different matter of course so I guess technically he could be trespassing when he's off the lead.
I doubt it really matters - just curious if you could enforce such a thing on a public footpath. I would have thought if it was a permissive path or on access land then the landowner could stipulate what they like within reason. I know areas of Northumbria are completely dog free for example.