It's similar to English which uses "an" instead of "a", but it happens very very very rarely.
(Needless to say, "águila" and "ala" ARE feminine nouns, what's changing here is the determiner so the two a's don't clash, not the gender of the nouns)
But it's not that important, gender does add some redundancy and some error-correction to a language. It's not necessary (as shown by English) but it has some use.
French has quite a few words that break the apparent gender rules: un musée, un lycée, un mille (meaning "mile"; the homograph meaning "thousand" is feminine but usually doesn't take an article), le mort (dead person) vs. la mort (death), etc. All adding to the shit you gotta memorize. If you don't, your meaning will still come across but you'll sound "off".
(I don't even think I remember all my Vandertramp verbs...)