First of all, if you are telling staff that X is the policy, and they are giving patrons exceptions to X without clearing it with you – they need to be on a PIP. Now. I don’t care how nice they are, I don’t care what they do for the community. Libraries hate disciplining insubordination – but that’s what’s happening. If you have HR, get a standing meeting on their calendar about this ASAP.
Stop requesting feedback. Yes, good managers take feedback. Very great managers are actively soliciting it. There needs to be an established professional culture before you can open those channels though – the fact that staff are giving you feedback and expecting you to act on it, but are not doing what you tell them means you do not have a culture that’s ready for big picture feedback to a manager. That doesn’t mean you kick someone out of their office when they give you feedback, but stop actively asking for it yesterday.
Your focus in terms of your own improvement needs to be documentation. Keep notes from every meeting – have them posted in an accessible place. I inherited a particularly disengaged team once and made it a requirement that they had to send me three work-related things that they heard in the meeting before I posted the minutes a week later. Everyone hated it, but I told them clearly that if I knew they were paying attention in meetings and implementing the knowledge from meetings in their jobs, I would get rid of the requirement. Somehow, within a month, everyone knew what 95% of what was going on. (There was also a disciplinary conversation with the employee who said “oh, I figured I just wouldn’t do it.”) Keep a running bullet list for every employee – good things to praise them on, things to coach them on, things you’ve explicitly told them to do or not do. Set aside an hour every week to go through these notes and do more long-term thinking about what each employee needs. Also, if you have people who are generally doing the right thing, bring them on board. I had an employee that LOVED to make infographics on Canva. She made so many “here’s how to do X tech thing” half-sheets that staff embraced. This also helps with the boss vs everyone else dynamic.
For staff pushback. You need to internalize this – you are the boss, you get to make the final call because that’s what you are getting paid to do, and it is okay that people will disagree. If a staff member is giving you a lengthy essay about revamping the fine and fee schedule, you don’t have to get into the details. You don’t have to let them rant for 20 minutes about how how they hate the current policy. You can tell them, “you are raising your voice and haven’t offered a solution – please leave my office, we will revisit this another day.” This also sets a tone – “If you can succinctly explain the problem with a policy and a potential solution, Boss is always really receptive. If you’re rude and unprofessional – they won’t listen to you.”
You have likely got to get this staff under control before you can really do anything about the patrons, because your community knows that they can game the system and find a librarian that will give them what they want. A helpful staff training about how everyone is implicitly biased and more likely to bend the rules for people who are of the same identity can help reinforce why exceptions usually end up hurting the most vulnerable patrons.
You already have a lot of advice for the patrons, so I mainly focused on staffing (though one rule I have with patrons is if I have explained the policy to you once, I don’t explain it again. “You have been told before why your group isn’t eligible to book the room. Have a good day.”)
Don’t feel like a bad manager. They do not train us in library school on how to be a real manager, and I think the industry as a whole attracts people who don’t always have a good idea of how the average professional adult workspace operates, which enables some really toxic behavior like yelling at your boss about a policy change. I know a lot of my advice is basically “become a hardass” but being super helpful, kind and receptive to feedback are only a part of the recipe of good leadership. You are on the right path in that you know what’s going wrong and how it’s bad for you, the workplace and your community – it’s just gonna be some hard work at the start.