Personally I think the '4th trimester' thing is rubbish. By that I mean this idea that they behave a specific clingy way for 3 months and then they are ready to be a bit more independent.
I think the reality is that want to be near you because that's the safest place, and you are their safe place, rather than then being born too early.
Some babies will be able to sleep without contact immediately, some will take a few weeks or months, and some longer. There isn't a magical date that it happens, but it does eventually happen for all of them. Same with things like feeding to sleep (15m and 2.5 respectively here).
Personally, both of mine needed contact to go to sleep for years, but didnt always need it to stay asleep, so we could do sleepy transfers (than enjoyed snuggly naps where we could).
With our first we used to put her slightly on her side, at least at first. We'd also swaddle, use Ewan the sheep, and pre warm the cot. We also got a (lightweightish) wheat pillow that we'd put on her when we out her down to replicate the feeling of a hand. After about 5 minutes, we'd remove it and she'd usually stay asleep.
My second would go down as long as there was an exposed boob within touching distance, preferably with her hand on it 😂