Why can’t you, him, make it official?
It boils down to whether you care about it or not. Your posting flip flops on this. Posting in the first place, saying you were gutted you didn’t get engaged over Christmas, says you care. Saying you would stay in the relationship if you never got engaged/married says you don’t care about it, or reworded- that you do not place value on it.
IImho, your focus on other couples’ relationship progress is a mistake. They have nothing to do with you and their relationship is irrelevant to your relationship, iyswim. Your “baby daddy” (something worse than ‘boyfriend’ or ‘partner’, no?) may not want to feel manipulated into bowing to peer pressure.
Are you financially independent from your bloke? This is the crux of the matter. You need to be if you are not married.
Have you told him you will stay in the relationship regardless of being married or not? If so, I don’t see where you would have any leverage to “encourage” him to marry. You’ve moved in with sex on tap, have had and care for his children, and declared your everlasting loyalty: he has all of this without legally binding half his net worth to you. Why would he propose at this point?
I think he is stalling as well, sorry.