I am rarely sitting about doing nothing
One strategy is to match his 'sitting'. Just sit on the sofa. Sooner or later, something will need to be done (DC needing feeding or putting to bed etc.), at which point you ask him which of you is going to do it. If you get up, make a mental note and remind him it is his turn the next time something needs doing.
Repeat the next day, and the next, until the balance has shifted.
This way you are not attempting to measure who is doing what tasks, you are measuring how much 'time off' you each get - the latter is easier.
But you can only take this approach if you are prepared to let the housework and everything else go to pieces, so that only absolute essentials get done for a good long while. Could you bear to see that?
I am the ONLY one concerned about looking after finances (he won't even look at his bank account - he knows he's rubbish
Count that as a win. Far better to have all the control than to have no control and feel like you are being kept in the dark and potentially financially abused.
monitoring appointments for kids, planning anything for the future or even weekends.
It is very hard to drop the mental load where the kids health is concerned. There are no answers to that.
You can drop planning family activities. Do what you want to do with the kids, and leave him at home, but only do things that you yourself really want to do.
He completely checks out with his game on his phone [...] I feel like there is no adult conversation. I have spoken to him and asked him to be involved with things [...] by the time I get the kids settled
Why is it always you who settles the children? What would happen if you simply didn't get up off the sofa and told him to go and see to them?
he is on an ipad [...] Unless ofc we have spoken about having sex, when he will lie there waiting for me. There is nothing that makes it feel more like a chore after that.
You are conflating two different things here: wanting him to be involved with the chores, children, and family life on the one hand, and on the other wanting him to engage with you as an adult and put some effort into your relationship.
Either way, he is on his phone or iPad and checked out, but you need to be clear what the main issue is: do you want more practical input into the home, or are you lonely in the relationship and looking for more time together as adults instead of parents.
Have you lost your identity as an adult, separate from being a parent?
The bottom line is that you can't change him, it is not possible to make someone change. You can only change yourself. Match his energy; sit on the sofa beside him and do no more than he does.
If that is not what you want, then divorce may be the inevitable result