Just picking up on this - you wrote this post at 13:36 so it was well past lunchtime. I don't know if you were referring to the hypothetical "take them outside" suggestion from 8:30 or whether it was still the case that you had all those barriers between your situation and gettign them out at 1:30pm but it made me think about myself. I have ADHD and struggle with these things. The reason that people suggest things like "just" taking them out is that most people have routines and systems which mean that it is easy to just go out - they get up, washed/brushed, dressed etc first thing. Get DC breakfast and don't need to sort out a bag because they already have a bag ready with essentials like purse/keys/whatever and/or stuff like sun hats, water bottles etc ready accessible near the front door.
When my ADHD is getting on top of me and/or DH is ill/away or something, it's hard to do everything because nothing is "just" doing that one thing, there are at least 2 subtasks which need to be completed first and often more like 3-4. So even something like getting them breakfast is more difficult than it needs to be for example, because nobody has cleaned up the table from last night's dinner and/or the kitchen is a tip because nobody tidied up there. But when I am more on top of things generally and DH is on reasonable form, which TBF is most of the time these days, then everything just works smoothly - we can come downstairs and easily be able to grab plates/bowls, food, drinks etc, chop up fruit or whatever and bring them to the table and there is nothing in the way or making you feel disgusting like food crumbs on the floor or washing up staring at you from the previous day.
Oh and trust me I have done the separate screens in different rooms - it does buy you quiet in the short term but too much of it just makes them more demonic 🙈 100% recommend avoiding this as a solution, unfortunately, especially if you're close to burnout because when I'm burnt out, I find I can't use it sensibly e.g. for 20 mins at a time just to give everyone a breather - I can't bring myself to end the quiet, sometimes for hours. And then they are just nuts because they obviously have all this pent up energy and it's even harder to direct it into anything reasonable.
Secondly though 3 is just a complete bastard of an age. And the combination of siblings when the youngest is under 4 and/or the eldest is under 6 or so is really hectic and needs a lot of adult management. 
it is a really hard stage. It does get easier.
I hope this does not come across as "this is what's wrong" but for me I've noticed that everything can snowball really fast when I start getting behind on things because one thing slipping makes everything else harder and then I'm less likely to be up to speed on the next thing and in a very short time it just starts to feel impossible to keep up. Comparing that to how things are day to day when we are on top of things enough to be a couple of steps ahead of the DC is just night and day, and I think IME that's what makes it feel like advice which works for everyone else is impossible or doesn't work for me. And it is much harder when you're a single parent - I have been a single parent (with DC1) and because it's all on you it's like this is all exacerbated because there's nobody else to pick up the slack when you're having a hard time.
Being a couple of steps ahead of them means that I can do things like distract the younger one or involve them in some activity so they aren't bored and looking to wind up their sibling. The hair pulling sounds extremely infuriating and I don't know if I could react to that calmly in any situation.