Why does anyone think that now the space is shared with pedestrians the riders should have to conform?
Well, I don't think riders should have to conform. However I think it would be sensible for riders to consider that if there is not a bit of give and take, there is a very real risk of further limitations being imposed on equestrian access. If a path is heavily used by pedestrians, narrow, and fairly urban - like a tow path - it is possible for its status to be altered - from bridlepath / open access to pedestrian only footpath. That's a massive loss for the horse riders who use it.
I think it's sensible to try to avoid that situation by not leaving droppings all over the path, or by returning to remove them after riding. Even if that's not always possible, it's better than the "annoyance" factor to other users is reduced as far as possible.
Similarly, there is no reason for horses to be on pavements, other than in exceptional circumstances (literally, avoiding an imminent accident). If I put one of my horses on a pavement and it left droppings, I would try to swing by and pick it up on my way home from the yard.
On a road, [mud-type] bridleway, or similar, I have never even considered picking it up. Even where that bridleway is a narrow path on which pedestrians also walk - because it's a bridleway and pedestrians should anticipate encountering a horsey presence there.