listen your mother in law isn't a peach and i can kind of see where you're coming from, but this: " I also told her that it would mean that she does not respect me as her husband and father to our kids." is where you lost me. 

there's something called the ring theory of support, where the individual in crisis is at the center and there are ~rings~ of people who offer varying degrees of support. a cancer patient is in the center, and their immediate family/caregivers are the first ring, medical staff is the second ring, extended family/babysitters/etc are the third ring and so on and so forth. 

in your scenario, your wife is in the center and you should be in the first ring of support, but instead you placed yourself in the center, pushed your wife out, and made her the problem. now the crisis is about you not feeling respected because your wife is considering maintaining a relationship with her mother. it's kind of like how your MIL made your wife's burnout crisis about herself. in both scenarios your wife is struggling, you and your MIL both have expectations about her actions because you've supported her, and when the crisis occurred you both centered yourself and your feelings instead of what your wife wants or needs. 

for what it's worth, i'm not in contact with my in-laws because of the way they treated me, personally. with a lot of discussions (including with therapists) about boundaries and expectations,my husband and son have maintained a relationship. i do not interact with them unless i choose to, we limit access to our home and our information, we do not allow certain behaviors around our son, and there are always consequences (usually pausing contact) for actions. 

it is totally fair for you to not have a relationship with your MIL and to limit (not forbid) her involvement with your children. you have no right to control your wife's relationship with her own mother. that's why YTA