Mental illness occurs when it is triggered by things. Sometimes this is a chemical process, but often it is biopyschosocial, by which I mean it is caused by multiple factors.
Sleep deprivation affects all three of them, the biology of the body, the psychology of the individual and the socialogical factors in their life can all in turn become triggers for a mental illness. Ignoring them all repeatedly to care for a baby is not the best idea unless you have very good coping techniques and lots of support.
For me sleep deprivation caused a paranoid episode, verging on psychosis, and it took a long time and a lot of medication to get that back under control. Lack of sleep is now one of my recognised triggers for poor mental health.
Different people cope differently, you cannot sweep aside women suffering the side effects of sleep deprivation and say we must knuckle down, it's our job as mothers, and also excuse women who have issues with mental health, because often one leads to the other.
Our capacities for coping with different things differ, and as a mother I was the kind who valued the opportunity to rest as well as my baby be cared for because for me, that method best gave us what we both needed most at that time. What was important to me, after establishing breastfeeding and lots of skin to skin time, was protecting my mental and physical health, so that I could continue to care for my baby the best I could and leave hospital in good health.
I don't have a problem with anyone else having different priorities to me, or different approaches even to solve the same problem, but I don't accept anyone telling me I did it the wrong way, when actually I did what felt was optimal co-caring for me and my DD.
Mother and child is a relationship, both of their well being is important in raising a child well, sub-standard care for mother can result in mother giving sub-standard care to child, and I firmly believed that I had a duty to myself and my child to care for us both through this.
As a disabled person also, I recognise that when I don't look after myself, I end up less able to care for my children, and whilst burning as a martyr might keep my children warm for a while, when the fire goes out, they'll be cold AND alone, metaphorically speaking, so I had to learn to say - actually I'm ill/not coping/deathly tired and I need some HELP thanks. Obviously some people may be in a similar situation and breeze along fine and need nobody, but that doesn't invalidate the fact that some people won't.