Hi OP, I really feel for you and think that it's hard to make it work for everyone in a family, so often the burden to keep everything going falls to one person, and it's exhausting. We've been doing this for a while now. My kids are 9 and 7, my husband is an equity partner in a city law firm and I am a professional (not a lawyer, running a small private practice). While I can't purport any expertise as I do sometimes feel tired, old, burntout, chaotic and in the middle of an unremitting shitstorm, DH and I have tried to learn from our mistakes:
The couples I know who manage this best, both take responsibility for family life, and have an equal investment in making it work. That does not necessarily mean doing an equal amount all the time, but that no grown-up gets an opt out, ever. As part of my professional work, I see families when life isn't going so well. One of the interesting dichotomies I've noticed it that of families who are essentially 'together' in supporting an encouraging each other and families who have a less cohesive approach (competitive tiredness etc.). I'm not making a judgement about whether that applies to you, but I'm impressed by how some friends have been able to make the seemingly impossible work by intentionally creating the 'together' approach.
DH has negotiated flexible working. He works from home a day a week, does two school runs a week ( so I can start early) and we split events such as sports day and carol services. He's done this ever since he was a junior partner and it's in no way affected his career. Obviously sometimes he has to be flexible about going into work or travelling, but most weeks it works fine. He is lucky to work at a very forward thinking firm, and as he's got more senior it's become easier if anything and over that time the work culture has changed to make this not that unusual, and he's been really active in making that happen at a firm-wide level.
Outsource, outsource, outsource. After school club is fine a couple of times a week. We have a cleaner who also does our laundry and a couple of guys who do the garden when needed. Recently we've cut back on a few clubs as it was all getting to much. My children seemed almost relieved about it, and we really love just hanging out at home in the evenings and it's been a revelation to get some of the weekend back.
On a practical level, we have a shared online family calendar, all school emails go to both of us and we both have access to the Amazon and Ocado accounts, so shopping can be done when needed. I do more of the 'life admin' at the moment, especially around my son's SEN and my husband notices that and says thanks, which I appreciate.
Take time for yourself and let your husband do the same. take a weekend to go to the theatre with a friend, or to go for a run or to a yoga class. And take time to do things as a couple too. There can be so much focus on ensuring the kids have the very best activities and stuff, but everyone in the family should be allowed to get the best out of life.
Wishing you the very best of luck OP, I know it's not easy and there's no right way, just what is right for your family.