I agree with everyone else - this isn't negative feedback, your team have been asked what could be improved and they have said this.
You sound massively overworked and stressed, to have reacted like this.
I've been in your position - manager, huge public sector organisation, overworked, not taking lunch, I was always the one late for my kids picking up from school because of one last phone call or email to answer. I considered myself hardworking and conscientious. I had colleagues at a similar grade who did f all, or spent their entire time adding to their cv.
I ended up really unwell with neck and back issues because of the huge workload, glued to my desk and computer. My theory is that people allocating work will dump it on the most conscientious when everyone else has slopey shoulders.
Then an incident happened - I had reported serious wrongdoing by a colleague (I was duty bound to), and my corrupt, misogynistic manager decided to shoot the messenger. I'll not go into it all, but I ended up going off on ill-health retirement due to his harassment.
I now look back, and I was absolutely crazy to prioritise work to the extent I did. I thought I was making a difference to people's lives, and I suppose I did, but at what personal cost? I also think that one of the reasons I reacted so badly to what my manager had done (it was objectively awful what he did though, and I ended up with PTSD) was that I had poured so much of myself emotionally into the job. I now look back and feel angry with myself for not having better boundaries.
My advice is to try to use this as a reset. Push back on your manager and have a realistic discussion about workload. What can you stop doing, or delegate? In my role there was so much crap that 'needed' to be done, but in fact the world wouldn't stop if it wasn't. Stop going to meetings, or send someone else, or suggest they take place less frequently or do them remotely.
I would also knock making an effort around birthdays on the head. It's not your role, it makes you look desperate to be liked and it weakens your position as a manager, especially if you have to start performance managing someone.
I really hope you are able to lick your wounds over the weekend and start to look at this objectively. Your team want you to be more accessible for them. That's fine. Make sure you are, but make sure you start putting in boundaries around your own workload.
It might also be an opportunity for you to give some thought as to whether this is an organisation you want to continue with. It sounds overworked and a bit toxic. Why hasn't your own manager been supporting you more? You sound burnt out. When I look back, I thank my lucky stars I left, even though I hated the way it happened.