1-minute time out on the stairs seems like a vacation?
He should be going to his room and he should either be put in his crib or in a playpen for his time outs. Let him sit there for 20 minutes if that’s what it takes. Do not pick him up until he has calmed down. Tell him "no hitting" and put him straight in his (play pen/crib), when you come and get him say no more hitting, we are gentle to our friends. Give him a big hug and be constant.
I don’t believe in time outs, they only work for seriously 10% of kids. I have a daycare girl who when put in her room thinks its funny & the stairs is a vacation as well. I through trial and error found that putting her in the corner works like a charm.
He may need the de stress of being isolated and after being removed from the fun a few dozen times it should really work. It worked for the most aggressive child I had in my care and now she is amazing at maintaining her composure when angry. If someone else takes a toy from her she is extremely calm. But boy did she ever hurt the other kids just out of the blue for no reason.
It must really suck that its your own little one but stay firm consistent and always hug after he has been removed from the group for time outs. When you are able to re direct him praise him for great listening. Remember it’s just a phase.
Good Luck!
Hugs