I May be in the minority but i hate dogs being on the beach. I hate that in this country people take their dogs to beauty spots to use as a toilet. Even if you pick up after it's still filthy and children will be playing there touching the sand, people picnicking there. It's gross
In my experience (ddog died earlier this year at the age of 14) most popular beaches have at the very least, a dog-free zone from May to September. It's usually in the most popular/accessible part of the beach. That's fine in my opinion - it was no problem at all for us to take the dog further along a beach to avoid the dog-free area.
Re. picking up - on a beach, sand will stick to the poo where it's touched it, and you tend to scoop up a handful of surrounding sand along with the poo so nothing is left filthy.
In any case, birds and wild animals will also poo on the beach. If you're out and about in any public place you might touch a surface that, unknown to you, has been touched by something 'gross' - I would have thought it normal good hygiene to wash or sanitise your hands before starting a picnic.