Looking at the list, it seems that you are a logical and systematic person who enjoys organising things. I may be wrong but it appears that you enjoy being "task focused" rather than "people focused", and enjoy producing a "work product".
I think the "serve, care for" seems to be a bit of a red herring. I'm sure you're good at that but maybe it shouldn't be the absolute focus of your job search.
I could be wrong, but it's because it's how I am. I'm a lawyer but with a science background, Before I retrained in law, I worked in the public sector for a while and I hated it for the same reasons you list (not enough budget to actually do anything meaningful, our "achievements" felt false and contrived to be for that reason). I never felt like I produced anything.
In my job now, for example, I analyse documents and "produce" a written piece of advice. In doing this, I help my client with what they should do next, and they are usually pleased with my work overall. So that is where a "serve" element comes in, but obviously I'm not a nurse or a waitress so it's not "serve, or caring for" people directly. Do you see what I mean?
I should add that I career changed when I was 29 (I'm 37 now). I really enjoy what I do and I'm so glad I did it.
I can also recommend the "colour personality test" for finding out whether you're introverted or extroverted, and task focused or people focused".
In terms of finding a new career, I find it easier to think about what tasks I would like to do day to day, rather than the overall picture of what the organisation achieves, to find fulfilment. Although I thought that I would find working for public sector fulfilling because the cause was good, I didn't, because I didn't enjoy the tasks of my job. My job in law now is probably less "for the greater good" but it helps my clients, and they are trying to do good. Hope that makes sense...!?
Good luck!