“It says in my ArtScroll Machzor that you’re not supposed to daven for personal needs on Rosh HaShanah.”
That’s not exactly true:
1- You do say “Elokai Netzor” which is quite personal.
2- One can – and should – ask for spiritual things <i> and anything necessary for spiritual developement.</i>
3- Considering that most people don’t sincerely feel the strong love for HKB”H that precedes wanting to be Mamlich Him, it’s better to have kavanah for something than fake yourself into believing you really want to be Mamlich Hashem (when what you really want probably is at best for the galus to end so you have it easier for whatever reason, whether physical or spiritual).