I'm okay with sadness, and being shy, but screaming and sobbing is the part that seems extreme for the age?
It may seem extreme but perhaps it isn't for this particular child. It may be that he has separation anxiety and that isn't something that is going to go away easily at age 4 and will take some additional work for you outside "waiting it out" and a regular adjustment period.
I advise that you work collaboratively with the parents and establish patterns of behaviour at home and how parents respond to certain things. This will give you a better idea of what negative behaviour is being reinforced rather than making assumptions and getting frustrated. Aside from the crying is this child well behaved? If so why not have a space where the child can be alone but in your sight, with comforting items as long as he is calm and quiet and then he can join the group when he is ready to join in and be happy. He is old enough that you should be able to communicate with him enough to find out what he needs from you to be happy in care. I imagine once he has settled in the younger sibling will follow suit. Have you got a play tent you can put up so he can go in there and play cars or read books alone till he cheers up. You are offering him a safe space that way where he is allowed to feel what he feels but at the same time letting him know that outside of that space he must not sob and cry as it is disruptive to the group. Whenever he feels sad he can go into the tent, or space, and then come out when he feels better. Eventually he will not need it anymore.
It kind of sounds like the child is overwhelmed by the sudden change in life. And, if his sibling is not coping well either then it would make his adjustment harder. At 4yrs old if you are stressed about something, having another child bawling and stressed about it doesn't help you build confidence!
I think that as hard as it is for a 12month old to adjust to daycare it can be even harder for an older child. 12 months old don't really have a clue whats going on and just get used to the new norm...4yr olds can sit there all day remembering what their life was and comparing and grieving the loss of old ways etc. While it is harder to have patience for them at that age we still need to be put in their shoes and figure out what they might be reacting too and try to work with it.
I'd suggest identifying the emotions. "it's hard to not be at home anymore, it's hard to not have mom with you all day" etc etc. Give warnings before you transition. Transitions are really hard on kids that are not settled and feeling safe and confident. Give lots of warnings as to what is to come so they aren't taken by surprise.