Mother-in-law exacts emotional toll for her child care services
However, your wife doesn't shrug this off; she's actually overruling her own judgment in response to the criticism. Even if she doesn't agree with Milly's take, she's buckling under the weight of it and making choices she wouldn't otherwise make.
That's poison. It's bad for your marriage, you're dead right about that -- but it's also corrosive to your wife's confidence as spouse and parent, and confusing for your daughters. Let's not pretend you're keeping it from them, because kids learn to read you fluently well before they can form the words to explain that.
Not to mention, this dynamic is flat-out terrible for whatever relationship you two have left with Milly.
That's the least of my concerns here, since Milly seems intent on smashing that up single-handedly, but it might hold sway with your wife -- and you need her to summon her strength to resist Milly.
To see how, let's zoom in on two modifiers in your letter: Milly provides day care, but to do so "graciously" would require some baseline respect for you two as parents; and day care is "practically" unaffordable, which means you can afford it, if barely. Yes?
These words are your two points of access to the problem. You either set whatever limits you must with this caregiver for her attitude to pass for grace, or you curb your spending enough to afford a new caregiver. It is really as simple as that.
The former is profoundly overdue and best handled by your wife, but if she's not up to it (yet? See below) then it'll suffice for you to handle it with your wife's blessing. Here's how, using your example: When Milly says it's "selfish and self-centered" to go out, you say, "We will hire a sitter then. Please do not tell us how to conduct our marriage."
Period-full stop-done done done.
When Milly lectures your wife, stand up for her where she can't stand up for herself: "[Wife] is an excellent parent. I'm grateful for her every day."
When Milly persists: "You raised her. Maybe it's time to trust you did it right."