Im not sure what you're missing here, because from my experience I'm always out level my progression quest and my gear always outdated because my leveling are faster that it should be.
I'm able to ride my chocobo at lvl 25(supposed to be lvl 20) and my access to Sastasha at lvl 20 (supposed to be lvl 15). What Ii'm done is doing all nearby quest and queue dungeon, not even touch the hunt log because at that time I didn't know that thing is exist.