Typically I arrive home after work and offer the shoulder to cry on / emotional support, sort washing, wash up, help with dinner prep (I arrive to late to take on full responsibility for cooking though). I have full bathing responsibilities (which is now my time in the week that I get to spend with the children) and try to encourage DW to rest while I do this before putting DS to bed. We have an arrangement where she takes off one day a fortnight away from the children and cooking. I could in theory do a bit more I suppose. When we're both at home I aim for at least 50-50 with some added headspace for DW.
Christ, you sound marvellous, where did you come from?!!?!?
Seriously though, it sounds as though there are two distinct parts to what's going on. You know that DW is having a hard time, and is shattered, if not suffering medical fatigue. She needs to get to the doctor but is having a hard time making it happen. So help her. Ask her when she can go. Does she need you to take time off work so she can go without DC? Is she afraid to go? see if you can talk to her about it. If she doesn't want to talk, get home early, make dinner, put her in a hot bath, then try. Has she had a touch - or more - PND, or depression? Is she happy with the balance in her life? would she rather be at work than at home? understand that motherhood wrecks your body, seriously messes with your sanity, and annihilates your identity and self esteem - for some women this lasts a few weeks. For many/most, a lot longer. It may take her a few years to come out the other side. People who feel tired and crap about themselves don't want to engage in intimacy. They don't want to feel vulnerable and exposed.
Only once she is well in herself can you really address the question of sex in your marriage. And the short answer for me is, without sex, it aint a marriage (because sex brings intimacy, which I think can't be compensated for in other ways). And to keep a sex life up, you have to have a lot of I-don't-really-feel-like-it-but-let's-have-a-go sex. That is NOT anywhere near not consenting. It is thinking I would like to have sex with this person in an ideal world, but I'm so fucking tired/cross/fat/whatever that I won't. And then thinking it doesn't matter I'm tired/cross/fat/whatever, once we get going I'll forget that.
Other people feel differently and that's ok - unless two people in the same marriage see it differently. (FastLoris said it better.) You need to be honest with DW about your need for sex and how closely it is bound up with your identity and self esteem.
Do you still fancy your wife? love her? does she know that? are you sure? Unfortunately with most things in life wanting to receive more usually means giving more first.
Finally, like anything, it's a habit, and if you get out of the habit, it can seem like a mountain to climb. Reintroduce sex gently, start touching her more without expecting anything in return. Foot and head rubs, build from there.
And yeah, learn to be happy with less. Sorry