That's a significant age gap, but ateotd a man can have kids almost indefinitely.
(Yes, past 39, mens fertility is affected too, but they can still have them in a way woman can't post menopause).
So it made no difference to your Mum, because she still had the opportunity to have kids.
(If it made a difference in other ways, it was clearly not enough to cause her to end the relationship).
He is acting very much like someone who wants out but who is flip flopping all over the place and also wants to see what/who he can get and set something up before he gets out.
I think, given the chance/opportunity, he'll keep you going until he feels he has forged a viable, solid option with another woman.
This approach, and the lying while swearing on his kid's life (and, as someone phrased it well, the weaponisation of his therapy) are all reasons you should he glad he's looking to get out, avd will.ve someone else's problem. Because those are serious, serious character issues.
A decent man could have perhaps ended up in this position; but he'd be honest/straight and leave; not do what he's doing, going on dating sites behind you back, lying, swearing yes not lying on his child's life, and giving you a contradictory, to and fro, self indulgent, BS narrative that is self serving. Self serving because he's not ready to jump yet and might have some mixed feelings.
Did he get serious with you and move in together (and then co-buy) partly because he wanted an easy, secure, soft landing solution to his living etc circumstances after the breakdown of his relationship with the mother of his child?
It all seems very fast/rushed and careless.