Currently transitioning from one job to another, but both senior management roles - don't really want to be too specific.
DD is a teenager now, so thankfully no more night wakings 
but she was terrible for the first few years! As things worked out, I actually did all of the getting up in the night with her, because I breastfed till she was nearly three, and we just fell into that pattern. I do remember how hard it was to function on so little sleep, so you have my sympathy.
I think childcare/cooking should be shared at the weekend, but if one person is at home all through the week, there shouldn't really be loads of other stuff that needs doing.
I actually did a split shift when dd was little, so I worked mornings and evenings and then had the afternoons from 1-7 with my dd, doing housework etc. I regarded that time as my "chill" time, it was definitely easier than being in work. But we're all different and all children are different, so I accept that the experience won't be the same for everyone. I just think that, if I was finding it such hard work to be at home, I'd be considering whether a return to work was the better option!
Of course, if you have a partner who wouldn't step up and share the load if you went back to work, you have a different problem entirely. I'm talking about how I would have felt if my DH - as the lower earner in our relationship - had decided that he wanted to be a SAHP. I preferred for both of us to share the load at work and at home, and would only have been willing to support him staying at home if that had removed the burden of housework from me. All of the benefit would have been to him otherwise - more time with dd, more relaxed lifestyle, freedom to shape his own time - and that wouldn't have felt like a fair deal to me.
Not trying to denigrate what you do in any way, I get that being a SAHP has its challenges. I'm just trying to explain honestly how I'd have felt in that situation.