First thing to say...although you are probably sick of people telling you this...it is just a phase.
I think it is quite easy for an idyllic bedtime routine which worked well when they were toddler/baby to get overextended. So one story becomes three, bedtime chat becomes longer, bath becomes longer...bedtime turns into an all singing all dancing entertainment, which increases the likelhood they don't want you to go.
What worked for some friends was to make bedtime much simpler. No story. Bath at different time of day. Very straightforward brushing of teeth, putting clothes away, nightlight, kiss, moving around next door, adult voices close at hand, but that was it. Bedtime, day is over. They all slept on one level, I wonder whether that made a difference, no sense of being abandoned, or parents going downstairs.
What worked for us was, starting bedtime earlier. (We had an elaborate bedtime routine with two three year olds(twins) and one five year old
This made the panic when they didn't settle less ovewhelming. Also accepting that the old night night wasn't going to work. I had a long time of dd screaming when I left her, and her endlessly getting out of bed etc. I completely changed my approach, and lay down with her, and cuddled her to sleep when she was 4/5. So easy, and she went to sleep very easily. That was all that was required, just not leaving! She never woke in night. She got overtired easily, and was the sort who became hysterical the firmer you got. I tried the strict method for long enough to realise it wasn't working. She needed closeness, not closure!
My 3 year old son was different. He needed a very sensible firm routine at bedtime, a good story, a cuddle, and just to be left. Light on landing always. A book/(soft toys to play with) in his bed to look at at to wind down by himself. He slept immediately.
Different children sometimes need different things.
My first son went to bed quite easily at 3, but he had loads of attention from us and felt very secure at that age, so again a different personality.