I would never advocate leaving a dog in a muzzle with a stool guard in it all day, but a regular basket muzzle is absolutely fine.
My two are muzzled during the day because Sagan is a chewer, so I worry about him swallowing something that could hurt him. And I wouldn't want to muzzle just one of them - especially since Merlin rules the household. If they ever got into a fight, Sagan would not be able to defend himself in the same way if he was the only one with a muzzle on. They don't care at all and never make a fuss - in fact they know they get a cookie when I leave, so they can't wait for me to put their muzzles on them and then get their cookie.
The can sleep, eat, drink with it on. Sounds like your dog-walker is unfamiliar with greyhounds and has not seen a basket muzzle before. A basket muzzle is very, very different from the muzzle you usually fit on other dogs, and greyhounds are used to them. There is nothing harmful about it, and they can drink and pant with no problem whatsoever.
The only problem I'm having at the moment is that Sagan has a scratch on the right side of his nose, I think it's from his muzzle, but I am just going to put a soft lining in it and I'm sure he'll be fine. His fur is also dry right now so that's probably not helping.
If he hasn't worn it in a while, you might want to put it on him for a few minutes at first, maybe with a little bit of peanut butter spread on it, and then take it off. Then put it on for a longer period of time, etc., until he is used to it.