Maybe it is natural to a degree, but it still puts the MIL at a disadvantage. She shouldn't have to not care about her son as much as she always has, because there's another woman in his life. That's not expected of the mother of a daughter.
Aside of the fact that I couldn't stand to be in the same country as my late MIL. It's not that. I had DCs in my mid 30s and have always had a really close relationship with my mum. It was totally natural that i spoke to her a LOT more (and would have even if i hadn't loathed my MIL) than to a woman who is only in my life because i married her son.
When my mum visited each time after the birth of a DC of course she was more concerned about me than my MIL would be, why should a MIL be over bothered about the health of a woman she's only related to by marriage? unless they had a really close relationship before DC arrived.
I see it with my DH, he's not close to his family and barely facilitates contact. I don't care because i don't like my In Laws that much, we have zero in common and i wouldn't choose to socialise with them if we weren't related. I'm pretty sure they feel the same about me so we're polite if we meet and that's it. I don't lose sleep over it.
My poor parents, though, with my DBs kids. They do all the running, always have. He's only really in contact when he wants/needs something. But they have always stepped up, looking after his kids when asked (for a week at a time sometimes) but do they get visits on their birthdays? easter? mother's day (bro is always at his MILs for that, his wife insists and he goes along with it. I don't blame my SIL for that, except that my parents are totes lovely and it makes them sad. It is up to my DB to facilitate a relationship)
That's just how some families are. But it is bizarre to think that a labouring woman or a woman who has just given birth is going to parcel out her time scrupulously 50/50 between her own parents and the ILs. Unless she really really wants to
And i think that's how the Grandparent relationship develops.