Op my son is 4.5 and although we never used time out/naughty step at around 3ish he would take himself off to another room when angry/upset. At first we'd check in on him but this made him more upset so we started saying "ok, you're upset, it's ok to need some space. We're here whenever you're ready/need us" or something variant on that.
At first he'd kick off for like half an hour but now it's literally minutes, some days he doesn't even get up the stairs before he's calmed down (unless he's really upset/disregulated)
The difference is, this is his natural way of coping. We never used it as a shame/punishment and we always let him know we're here for him.
Leaving a 3 year old crying outside for almost half an hour is really not good. He'll sense your attitude of "being sick of dealing with him" and tbh your update of "maybe it was too long, he should have calmed down sooner" totally misses the point - he's 3 it's YOUR job to stay calm and manage him, not his job to sense mum is sick of him and too busy so he's got to stay outside the house.
Also, you say "appropriate clothing"... Was it raining?? If you keep doing this, what will you do in winter? I know I sound precious but it's easy to get into a rut and do the same thing to "solve" issues so honestly, what will you do when it's too wet or cold to stick your screaming child outside? You need a better way of dealing with it.
Does his dad wear noise cancelling headphones when he's sleeping in the day? My friends husband does this and honestly he'd sleep through anything with them on and the door locked! Might be worth looking into as even minus the tantrums it can't be easy with 3 kids to juggle knowing he's trying to sleep.