Is this the only time he cries like this? If he tantrums at other times, discipline will make it stop and a warning will apply for school as well. Part of the hard first daycare years is to prepare kids for how to behave for school-including when mom leaves. 3 isn't too young, my kids start at 3.
My friend's son screams like this if she tries to put him in any classes, swimming lessons, gym daycare, preschool, but he screams whenever he doesn't want to do something, and she's never discipline him, just ignored him and left places if he freaks out, so of course when he wants to leave a place he freaks out.
If it's the ONLY time he does it, and you feel you shouldn't discipline it, and it stops after you leave, sounds like he has to get tired of it on his own, and he will eventually. What does he get when he cries? Al little extra hugs and attention? Try to do less if you're somehow rewarding this or dragging it out. A nice confident cheery good bye and leave. I've heard my daughter's daycare teacher coach parents to "Just leave please" in different creative ways when the kids are crying every day and they stick around comforting them, because the kids always stop the second the parent is out the door.
If you feel you can discipline him, I would use a combination of, "If you scream for school x (negative consequence) will happen" and follow through (must be something immediate-he's too young to have it happen after school and make the connection) and also have an "if you go nicely today, we'll have fun at the park after" etc. Just the reward scenario alone won't work though if there is no down side to the wrong action.
It seems there has to be a way for you to do this therapy yourself-it seems odd they can MAKE people take kids to certain places...but honestly, if speech is his only issue, you are doing the right thing by getting him this help and it won't hurt him to learn the routine and get socialized while he's at it. The crying wont' damage him, and the learning to overcome the fear will be a good skill for him. Good work doing it and Good luck!