Tantrums are par for the course at three but she does sound like she could do with a firmer hand.
Consequences are difficult at 3 yo because they need to be immediate.
So screaming/rudeness at the party - I would have immediately removed her for a discussion about behaviour and only brought get back in when she'd stopped screaming and apologised.
Otherwise at that age a firm talking to can be very effective. They naturally want your approval and seeing that Mummy is annoyed/disappointed can be enough to reinforce the point.
This doesn't mean shouting. It means eye-level chat, with a low very firm voice. I always found that removing the room/taking them aside was very effective. Eg "x behaviour is not appropriate/unkind/rude/unsafe etc. It is not acceptable. I love you but am disappointed with your behaviour just now because I know that you can behave very nicely. If you stop crying and promise to behave nicely we can go back inside. Otherwise we'll have to leave. Please say sorry . "
The hug is important because sometimes at that age they get a bit overwhelmed by the power of their own emotions.
Gentle parenting is all very well but you have to parent for the child you actually have. It sounds like you have a strong willed little girl (which is excellent) but her life will be happier if she learns to behave better.
Three and four can be a difficult year but it's import to instill discipline now before school. You have a responsibility to your daughter in this regard.
In terms of gentle parenting. My experience is that the families with the strongest boundaries and clearest expectations of good behaviour do the least shouting.
Be firm, be consistent, never give in to tantrums.