I think THIS is the real issue. He’s not so busy that he can’t take his plate from the table and put it in the dishwasher. That’s a one minute job.
the reason you are seething about the situation is that fundamentally he is acting like he is too important to do even the most trivial menial task and just expects you to do it for him. You need to talk to him about this. This is just basic disrespect for you. If you don’t address the real issue, the resentment will kill your relationship.
it is one thing for you to be responsible for managing the house/housework/childcare, it’s a completely different thing that he thinks he is too good to put his own plates in the dishwasher and expects that you will do it for him.
getting a cook, cleaner, childminder, gardener, whatever will not solve this problem. You need to sit him down and tell him that he is lacking basic respect for you. This is why you feel like a skivvy.
My DH used to have a highly paid job where he worked a lot and was away a lot. Yes, I resented him for not doing his fair share at home, but he did not expect me to do even the most basic tasks for him. He still contributed to the family, cooking and cleaning the kitchen, taking the kids to daycare, etc. He is just now starting a new very highly paid job where he will be working and travelling a lot. I do expect a lot of what he has been doing to fall to me again but I won’t be clearing his dishes for him. My DH can’t do the laundry if he’s in another country, so of course I will have to do it. But when he is here, he will cook a couple of times a week, and clean the kitchen up afterwards. He can put a load in the washing machine and turn it on then put it in the dryer when it’s done. That is 10 minutes of effort total. He will also take the kids to school if he is not leaving for the office at 7am.
Your DH is taking the piss, that’s why you feel this way. Please have the conversation and come and tell us how it went. If you just get excuses why he can’t do anything, then you know he is taking the piss. If he agrees to do more and actually does it, then your marriage might be ok.