I don't think you can generalise as lots of women (and men) have fab mils. But my experience is very similar to what you describe op. To answer your questions:
Yes- ime my relationship with my mil changed dramatically with the arrival of dc. As it did with the other 2 dils. Fundamentally we get on well but she definitely changed with dc arrival.
My mil didn't finish school, married at 21 children at 22, no career. Although trained as a nanny briefly, has dedicated her life to raising her boys. Mothering is her 'thing' and her whole, entire life. Genuinely believes shes the best mother in the world. Frequently boasts about what a great mother she is, how well she raised her boys what a good job she's done etc etc. Can't let go of the 'parenting' role.
So yes, when her dil had children she desperately wanted to be 'mothering top dog' Got very jealous and competitive.
Yes since having dc, my mil wants to keep me 'below' her- expects me to submit to her. Doesn't view me as an equal, a woman, mother and wife myself. Wants to control and parent me. This sounds extreme but it's done under the guise of 'helping' and acts of huge generosity a lot of the time.
The insults and jibes seem to part if this ime- it's control. Little messages that she's a better mum, better wife and that she thinks she's more important to dh than i am .
No you should not keep quiet to keep the peace. Over the years my relationship has vastly improved due to me not accepting little digs and insults and working very hard to be in control of my family.
She is fantastic with the dc, and is a great grandmother so I make sure she spends lots of time with them. BUT any attempts to 'parent' the dc or me are stopped immediately.
Part of this is being very strong and also set very clear boundaries that confirmed she's the grandmother not the mother.
E.g I can't accept 'free' childcare for example. Babysitting ok, childcare isnt.
Constantly reaffirming she is a valued grandmother and not mother figure in our family has improved things without destroying the relationship. Getting your dh on board and supporting you as the mother of you family now is very important.
Apologies if this has already been covered- I haven't read the whole thread yet.