"My partner is lovely and loving ..."
"... Then it’s the constant shouting about how busy he is and he doesn’t have time for this, and about how I’m always nagging him and complaining about how he goes about things."
Only one of the above statements can be true - lovely and loving men do not call their partners nags nor do they shout at their partners.
"Sometimes I try to “delegate” a task- eg can you call a plumber, will you take responsibility to book this appointment etc. I always have to end up reminding him several times, he takes ages to do anything, he has to ask me several times what to say, and then he always fucks it up- (eg books a time we obviously can’t make, hires someone completely incompetent, fails to communicate properly so the job gets messed up). ... it always ends up being so much quicker for me to do everything myself ..."
Sounds like Weaponised Incompetence to me. Where someone deliberately does such a bad job of a task that they don't get asked to do it again.
"he can barely hold himself together and get his work done , let alone contribute anything to broader family life."
More detail required on this, @Tandora. He holds down a job, so logically, he is competent at his job- as in he is able to complete the tasks of his job to his boss's specification. If he can do that, he should be able to participate in domestic duties too.
But on first reading, my advice would be to cut your losses and end the relationship. Feeling "unsupported and alone and drowning under the weight of my responsibilities" is no way to live, and you're already feeling resentful. It won't be long before resentment kills any residual feelings of love (if it hasn't already) and the very sight of him makes your skin crawl. It will be easier to co-parent if he moves out before that stage is reached. ~You're already doing everything anyway, it will probably be easier without him because you won't be having to do over anything you trusted him with (that he then royally fucked up). No shouting, no excuses, no fucking up, just calm and peaceful without him - a much better environment for your children to be raised in.