I have been in an uncannily similar position this year, I’m early 40s. I took a technical team supervisor position in a company where I should have been brilliant (I was overqualified) - I took the job as I was promised work-life balance and from day 1 that never panned out, the culture was long hours - just dumped rising volumes of work on us with no handover or planning and then expected my team to magically get it done. My line manager was caught in same trap working insanely hard - she left six months after hiring me.
i worked insanely hard work thinking it would get my team on track and we’d stabilise. Stupid of me. When my line manager left the department head assumed I’d pick up the slack as they hadn’t managed to recruit my new manager yet. Then, my new manager was hired with a different job title.
I flagged the workload and capacity issues over and over, recommended solutions, worked with IT and teams upstream to solve problems at source, automated processes and cross-trained and rationalised where I could, trained and delegated, spoke to line manager, spoke to HR manager, spoke to head of department. Basically I got told there was no budget for my (low-paid) team to do overtime and no bonus could be promised so my job to just push my team delivery harder and “automate tasks” (but no resource to do further automation and working with the worst systems I’ve seen in 20 years). Utter nonsense from senior management.
Very high turnover in my team due to workload and low pay. In the wider team also a handful of significant leavers and some people on short- or long-term absence due to work-related stress which sent more pressure on my team as jobs upstream being done wrong/late and we were stuck correcting their errors and making up for their missed deadlines.
Senior management simply repeating trite nonsense about work-life balance and family coming first whilst doing absolutely nothing practical to help. Final straw being humiliated and shouted at by my head of department on a management team call for not getting through the work fast enough (had two key people in my team out that month, one whose baby arrived 3 weeks early and another with Covid).
I felt very guilty about my sub-team of six people - i recruited them all and didn’t want to abandon them to such a difficult situation. And pride makes me want to succeed and not be someone everyone blames afterwards and says “she was hopeless, she left it in a mess”. When someone senior says “you should be able to get this done” I assume they have thought about it and figured out it’s possible. Because that’s what I do; I set realistic but challenging targets to grow and stretch my team. I have learned that most senior managers don’t do this. They set targets regardless of what is feasible and don’t listen to what is happening lower down the organisation.
So what I did was tell my line manager I was leaving and gave them a choice: I would either work my 4 weeks notice as normal (manic hours and effort) or I would work four months notice but I’d rapidly wind down my effort to my contractual hours.
My long notice was accepted as of course I knew it would be. Not much changed in month one while they mobilised a cover plan and I tried to work out how to ease down. Month two was horrible as I worked my contracted hours, the cover plan didn’t step up and I didn’t get the work done (got shouted at again). The last two months much better as emotionally I had got a lot of closure on the situation. The person who eventually stepped in to cover me clocked 250 hours of overtime in those first two months - he is charging them every penny of his extra effort which I honestly think is best way forward. With both of us there, we made some headway completing some system changes that reduced the work.
The irony being if they had found budget for 500 hours of skilled resource to work on the automation projects earlier in the year I would probably have stayed in my job!
So either way works OP - do what my manager did and bite your tongue then quit and be out fast in four weeks (“rip the plaster off”). Or hand notice in and feel some closure to reduce your guilt over “leaving your team in the lurch” and give yourself permission not to do the mad amounts of overtime expected.