Sorry OP. I'd be absolutely furious if my DC behaved in the way you described with the slide incident. What would you have done if that child was seriously injured?!
You need to get really firm.
He has clear boundaries at nursery.
He thinks he can get away with it with you.
How is he at home? Is he well behaved? Tackling behaviour at home is a lot easier and less embarrassing than outside with an audience of other parents. But imo, parents will judge you more for not parenting, than strict parenting.
Could he undo a pushchair buckle? If not, I get he is really too big but if he tips the pushchair over on the grass how hurt could he really get. He will only do it a couple of times.
You need to follow him everywhere - watch for triggers, intervene, get him to apologise. Try to stay calm, wait out the tantrum - try not to get to shouting as this is a reaction he is likely expecting.
Do you have a clear set of simple rules at home? Visable say on the fridge?
Lots of praise for good behaviour will also help.
Pushing boundaries is typical 3/4 year old behaviour. Nip it in the bud now. You might need to wait out lots of 40min tantrums but that time will pay dividends to you and your whole family in the long run. Don't give in.
It's a hard age. It's OK to feel exhausted by it.