If your hospital-doctor friends are single mums, then no, they can't opt out of picking children up etc.
In a hospital, there's a never-ending stream of new patients coming in. If you are a hospital doctor who does have a partner, I imagine it's very easy to say "Yes, I'll see one more patient" because said patient has been sitting in A&E for six hours and is in pain, and who knows, it could be really serious. I'd imagine many, many hospital doctors keep seeing one more patient if they can, until another two hours or however long has gone by.
There aren't enough doctors. OP, what's happening here is that you're effectively sacrificing your family life for the greater good of the nation! Only you know if you can make your peace with that reality. I don't expect the doctor shortage will be solved anytime soon.
I don't think that being a hospital doctor is compatible with family life, and I think most of them probably have partners who don't work or who have low-stress jobs.
If you want to stay with him, I second the advice here to get some kind of domestic help. Automate everything you can, like bills, and do online shopping. Batch-cook.
I know that none of this replaces his presence. And I hear you - while not the same, my exH was very much not present, for different reasons. It was extremely lonely.
Have you read Love Languages? I haven't, but the concept sounds good. Maybe your love language is Quality Time. If you decide to stay with him, maybe you could try to really make the most of your time together. Throw money at the chores and then throw money on doing things together, whether days out with DD or paying a babysitter and going on date with him.
He does sound lovely but also it's not nice being partnered yet on your own a lot.
Best of luck xxxx