We have had some of this with our younger DD, now 8. The tantrums were, I thought, the other side of "normal" since toddler age. Just - more extreme. To the point where a friend was once with her, on the way to collect dd1, and there were police at some roadside thing, and dd2 was screaming so much the police stopped our friend to make sure things were ok! (awkward.. apparently they were very nice though)
A friend suggested it might be anxiety, and it really helped to read about that; I talked to the SENCo at the school and she pointed me towards some ways we could help. She was not quite in need of extra help at school but her teachers had definitely noticed a pattern, iyswim.
It helps to notice if there are triggers, though they aren't always consistent. For us: hunger is one. Competition (especially competition that she is losing, competition with dd1) is another. Feeling 'hard done by' when asked to help clear the table, wanting to get a one up over her older sister, not wanting to help clean or get ready quickly or basically whatever we need -these were hard because it's hard to tell if she needs help in some way, or if she's just throwing a strop building to a tantrum to get out of tidying up. Much conflict ensued.
Things that helped -- for me and DH to get completely consistent; we were not, with me more the softie, and him more the 'she's trying it on to get out of this and we shouldn't tolerate it'. Helping her to articulate what she was feeling was good.
A big shift happened when I explained, several times, that we want her help because she is useful and strong and we wanted her on our team - this helps her to reframe away from "but why didn't [older sister] have to xyz right now too" to feeling more valued and understood. Praise for all things she does well and for when she is resilient in the face of something that would normally get to her is very helpful. Loves praise.
Her having set things that she knows are happening - cuddle time every single morning; this evening for dinner we are having xyz, then we are going to [whatever]; this weekend we will clean the flat, then go to blah blah, everyone is going to do their cleaning jobs - these help avoid some of the conflict points of suddenly being asked to clear something.
Good luck OP. Maybe boarding would be great for your family. Don't think it would have fixed our issues but I didn't have access to it anyway.