@PeachShark, it always fascinates me how so many Mumsnetters actually know why a stranger's MiL does or says what does. Do these all knowing Mumsnetters know exactly when a woman changes from being an individual female with all her faculties and sensibilities, into a MiL, and in this case a Grandmother too, who suddenly share all of the same characteristics and views on life - especially with relation to their DiLaws?
I wonder if it is as easy a transition as the second they become a MiL, or the second the baby's body leaves the protection of it's Mothers womb? As I am in my 60's, and both a MiL and a Granny, I presume that I unknowingly passed into this horrendous phase of life when at least one of those two (amazing, wonderful and very blessed) occurrences, happened.
I find it so sad to think that my DiL has to now deal with me, a nosy and bossy MiL, and Granny to her children, who's main aim in life (now that I have transitioned into one of "them") is to make my DiL as unhappy as possible, and to make sure that my Grandchildren always know who the most important adult in their lives is, and that I am the only one who knows how to comfort, and how to bring happiness in to their - my Grandchildren's - lives. I can only hope that none of my dearest of Granchildren "lose" me until the youngest of them is at least 20 years old!
OP, all my questions above are rhetorical ones, so please don't worry about finding an answer to them. Your MiL may be a complete cow, rather like the one I painted myself to be, but all of the PP's who are telling you exactly what and why your MiL is acting the way she does, do worry me that they are going through their lives with blinkers on. I do hope that they can learn to see things from more than one viewpoint or angle, which does of course make life quite a bit harder to manage, as one needs to put a lot more thought into any given situation...
Lastly, I do actually think that your MiL's words and actions might/could be coming from a good place. I don't/can't know whether they are or not, as I don't know her at all (well, as far as I am aware, I don't know her 🙄), so I can't know her sufficiently well enough to know the reasons for her words and deeds.
So, I will give an example of my own thoughts: My DiL is a brilliant Mum (and Wife to my Son, and DiL to me). Unfortunately, my DiL - like many first-time mums - is still almost costantly exhausted, just as I was with my first child, until she went to a fabulous nursery, and I went back to work. When I returned to work my DH no longer had any excuses for not doing his fair share of 'being the parent in charge of any night duties that cropped up'. My DD was also still breast fed at bed-time, or as a comfort feed during the day, but that was mainly for comfort - and a sweet way to get her sleepy, and also hopefully a lovely way to try to keep her belly reasonably full for a few hours at night-time!
Anyway, knowing that my DiL was exhausted for most of the time made me wish that I could give her a helping hand - well two hands really - but sadly my personal logistics made that impossible. However, if I was able to offer my help, I might well have tried to rock my DGC to sleep for daytime naps, so that my DiL would be able to trust me to look after her dear little one. So I think it is entirely possible that your MiL, Peach, was actually and genuinely, cack handily trying to help you.
Please don't take any Mumsnetters views - including mine of course - as being correct OP, as only someone who has known you all for a considerable length of time, would have any realistic chance of reaching the right conclusions, and/or suggesting reasonable courses of action for the near future.