Right, first off don't beat yourself up. You can only spread yourself so thin before you stretch too far and snap.
I have 4 dc - 12 (y7), 10(y5 and 8yo twins (y3). 10yo and 8yo dd both have adhd and dd2 also has issues working in the same space as anyone else as well as other battles round getting her to do work.
1st off, chances are the 'cant work with X' is a BS excuse to get round not being supervised. Ask what they would do in class if someone was distracting them - I'm pretty sure their teacher would remind them all to think about others but that they wouldn't be allowed to leave the class or use it as an excuse for no work unless there was significant sen at play.
Try the carrot and stick approach (works somewhat with my lsa students). For every 15 minutes of work they do, they get 5 minutes playing on a game. But if they take the mick when their 5 minutes is up, they lose that option until the next break. At home we have is completed by lunch, they can have x box in the afternoon. If they don't complete it, they lose the x box for the day and the work gets finished in the afternoon. If it doesn't, x box gets banned for the rest of the week.
With your eldest, choose whether to fight over doing work which should have been done and double his workload, or let him take the consequences of marking work done and continue from wherever they are now (week 7 in most places I guess).
Your dd needs to be praised for doing her work and the others need to see there is a reward for this, even if it's something small. - gaming time, sweets, tv choice - whatever works for you.
Your 8yo. Bribes are great, but sometimes they come to expect them. So maintain boundaries around getting a treat when work is done without the griping. What sort of thing does he talk non-stop about and is he easy enough to distract back to work?
The 6yo and their reading. Don't force it by expecting them to read instructions and apps. The text on a page can appear very different on a screen. But, if there is a book they like, I'd suggest a game of 'match the words' - so words you know they can read easily, write them out or highlight them on a page of text and get them to find them in onscreen instructions - it might be a confidence boost that they know that word and oh look, it's there as well.
Mine have tried every trick in the book and still do. But despite their tantrums, my rules are the same. No work = no gaming. Disturbing others or interrupting isn't tolerated and that child is reminded that they have to wait their turn. I try and split my time evenly but the truth is, 2 of mine do need more input so I praise the others for working well independently when they do.
They are also all vying for your attention because they aren't seeing friends, or socialising etc. The younger 2 could be encouraged to write to their school friends and relatives while you sit with the older ones. It doesn't have to be all about formal schooling.
But mostly, remember you are only human and you can only do what you can. The children will be ok as long as you can all be happy and relatively stress free.