Learning to fight as a human would have taken time, perhaps years, to accomplish, and Angel either could not, or would not, spend the time away from the action long enough to do so.
Being truly adept in magic, beyond the simplest of spells, apparently requires a certain natural ability in the Buffyverse that Angel doesn't seem to have, And again, proficiency would require at least as much time as acquiring fighting skills, and Angel doesn't want to wait that long.
"Fantastic detective skills?" His main contribution to Angel Detective Agency was giving orders and providing muscle. Wes, Fred, and Cordy did all the research and analysis.
Angel "mopes about like he can't fight evil without his powers, when he's surrounded by people who do just that," because, basically, it's the truth. As a human, Gunn could beat him easily in any sort of fight, simply because Gunn has learned to fight without the enhanced strength and speed of a vampire. At best, Angel hasn't had to depend only on human abilities since he was Liam, and we aren't even sure hoe skilled a fighter he was then. At best, he would be several centuries out of practice, and would have had little or no experience in any sort of martial arts training from those days.
Ultimately, it might (and I repeat, MIGHT) be a simple question of ego. Angel has been the alpha male in his world for so long that he would be, at best, reluctant to step down from that position and have to depend on others in a fight.