Well...aside from toilet training he sounds like a pretty good daycare kid...too scared to initiate anything on his own so just complies with you lol.

Sounds like there is a LOT of work to be done. Maybe sit down and write out all the steps you need him to master on his own to be fully independent and show it to the family. The child could very well be completely different at home so they don't realize how much time this takes.

Also...be very clear and direct in your wording. Telling him to go pee...to him could mean "go sit on the potty until you have peed" so no pee...he just stays there. "Go try to pee on the potty" might be a better wording for him. What happens if you leave him. Like more than 10min. I wonder if you leave him long enough if he'd do it himself. Or leave him but every 10minutes tell him from across the daycare "johnny, if you are done wipe your bum, pull up your pants and wash your hands" then leave him and repeat 10min later. At 3 years old I find it hard to believe he can't pull his pants up. Tell home NOT to send buttons or zippers only jogging pants until everything else is in place.

Shift the initiative to him. Stop helping...start coaching with words and then start putting distance in between you and the bathroom while you cue him with the words. "pants down, sit down, wipe bum...."

It should come along fast enough if he is doing it at home. Home needs to stop helping so much as well. As he gets more comfortable at your place it should help as well. 1 month of settling in with certain personalities might not be enough, especially if it is all being done for him at home.