My experience is less the usual take that “men do want kids just not with you” and more that with some men you need to strike while the iron is hot and that they are more amenable to things like marriage and babies in the earlier stages of a relationship, during the honeymoon phase.
I’ve known many men who are with their long term girlfriends say since uni then their girlfriend of 7 years starts talking about buying a house, getting married and trying for a baby. As a woman she needs to plan ahead if she wants kids due to a more limited window, fitting it around her career, getting married, buying a home etc. For a guy out who has been with the same girlfriend since his early 20s that might be the first real thought he has given it. He loves his girlfriend and is comfortable In the relationship but he also harbours fantasies of what he might still do, what he’d do if he were single. It is mostly all just idle fantasy but that and being out of the loved up, honeymoon phase means he feels trapped and pressured by talk of mortgages, marriage and babies at 28 or 30.
Sometimes the guy will in time accept that if he wants to stay in that relationship he will need to commit more seriously but other times the woman will dump the guy if he isn’t moving fast enough or the relationship is damaged by the pressure. Equally I know other women who stopped pushing for marriage and babies and then they get pregnant at some point anyway and everything works out fine. A uni friend of mine who is now “happily”married with two kids to his uni girlfriend was dumped by her one Christmas due to his failure to propose, he did eventually go crawling back to her with an engagement ring but not before he had tried to hit up me and no doubt other women to see if he might have a less pressured option elsewhere. I don’t think she knows he used that period to explore his options before going back to her.
My brother in law was in two long term relationships which were ended over pressures for marriage and children in his 20s and early 30’s. He then met a woman his age at 33 and within a year they were married with a baby on the way. It’s not like they have a fairytale marriage or that she was just magically the right person but she having been through it before and knowing she wanted kids and a husband moved quickly to lock things down while he was still loved up and amenable. Those who know them well know the marriage isn’t perfect or even all that great but from the outside, perhaps to his ex’s it looks like a textbook situation where he moved fast when he’d met the right person he did want kids and marriage with. That is just one example out of many I can think of.
There is probably also some truth to men finally settling down when they feel like their player days are over and they don’t want to end up alone so they finally settle and behave themselves, you certainly see this with a lot of rich or famous men who suddenly seem to meet the right one in their 50s, it’s perhaps less that the wife is loved so much more and more that temptation no longer abounds for them as it did in their younger days.