Do stop covering for him. Your email trail reflects his fantasy at the moment vs reality, which isn't ok. Especially as you're having to put a lot of effort into covering his gaps, mistakes and dropped balls.
Stop joining in, as it A. Makes you fuming! And B. Let's him succeed at presenting reality as super dad who needs to supervise the shirking and non reliable trailing mother.
You don't have to go nuclear or be rude to quietly but firmly refuse to join in this fantasy. At the moment it's a joint endeavour, him lording it over you and presenting himself as 'the man in charge'. You're letting him do this and propping his image up by quietly covering for him and stepping into the breach.
No wonder you're annoyed!
Push back, address things as they happen e.g. Thanks so much for X school papers, not sure if you realise but school gives us each a copy so I already have these, so you don't have to forwards them on, especially as he's so busy etc.
Or, yes I've organised 1 cake, what's the plan with his cake? And don't say you've bought them and explain how you've allocated your time etc! You've fallen into responding like a secretary would, and giving him an opening to organise your time and 'productivity'. Really not something you should be doing.
You don't have to enter into a tug of war with him, or a horrible power struggle. But you don't have to prop up his daddy fantasy and make him look good, whilst sacrificing yourself in the process. You also don't answer to him and shouldn't be making your private family life so transparent. It's your family life, and he doesn't get to know the nuts and bolts as it just gives him the power to order you around.
Gently refuse to engage every time he throws his crap at you. Either pass it back smiling and politely, (and ummovingly.), or just don't accept the shit in the first place.
Just step back and let it fall at your feet. Then he looks like an angry ape throwing clods of shit, just as he really is :)
'Oh let's keep this about the children'
'Let's focus on the present now'
'Right, let's get on with planning x now shall we?'
'I'll call back / mail back when you're able to talk about x, what about Tuesday?'
Or even maybe 'I hear what you're saying, but I'm not dwelling on the past/ I'm just focused on the present now, see you on xday...'
And so what if he whitters on about old relationship grievances? You've moved on and you don't have to catch that shit he's throwing!
You could start saying to others, 'oh dear, poor Richard (or whatever), he's very much stuck looking backwards, I worry about him' if you want to be annoying (!).
And as flr covering up his gaps in responsibility, don't do it! Say what is actually happening, without being mean, but being accurate, so others don't just hear his fantasy life. So 'oh I'm not sure why h isn't here' at the school meeting 'it's a shame, as we both had the dates'. Full stop. Move on.
Good luck, so frustrating for you!