Don't be so quick to discount Eorzean military technology and the apparent 'superiority' of Garlean machina - we ourselves after all shoot down a fully armed and armoured Garlean dreadnought airship during CM with a single ground-based anti-aircraft cannon (granted the cannon was also Garlean made, but that's beside the point) and the ship quickly becomes scrap metal after only a couple of hits.
The various wrecks seen in the Fringes are still on fire so their destruction was fairly recent, and was probably caused by Alliance cannons mounted on the Wall (which is very tall), so problems with range would have been reduced or eliminated.
To say nothing of research guilds like the Garlond Ironworks and the Skysteel Manufactory utilizing Garlean magitek concepts and adapting them for their own use, thus allowing the Alliance to effectively turn Garlemald's might against them.
Either way, don't write off the ability of the Alliance to bring down Garlean airships reasonably easily, as it's clear they indeed are more than capable of doing it.