That's generally an issue. You need to go past half of the tech tree, build some district(s) and build some new units. Moreover, AI rightfully prioritize Fighters for defense over bombers. So even though AI could use bombers, the possibility comes too late.
I think this illustrates a more general problem with the game: all the best units from the Tech Tree come too late in the Age to be of much use.
Mounted cavalry and full-fledged Hoplites and Legions - end of Antiquity
Arquebussiers, Bombards and Galleons - very late in Exploration
WWII Tanks, motorized artillery, monoplane fighters and heavy bombers (and aircraft carriers, but who bothers to build any of those?) - end of Modern Age, if you haven't ended the Age in a Victory by then.
My first couple of Military Victories were won almost entirely with Dreadnaughts and Landships - only built Battleships and WWII Tanks at the very end of the last battle, and frankly that was just because I could, not because I needed them for anything.
I would like to see some attention brought to the problem of 'packing' the end of the Tech Tree and the Age with the premier units of the Age, so that only a tiny fraction of any conflicts will actually be fought with them.
To put it in Antiquity perspective, both Legions and Hoplites are Iron Age units - starting 600 to 250 BCE or, ideally, available for at least the last 1/3 of the Age, so 'Iron Working as a Tech should come earlier in the Age to allow us to play with the goodies attached to it.
This, unfortunately, would require rethinking both the Tech Tree and the Units for all three Ages to be done right, so I don't see it happening any time soon, if ever (outside of some Gargantuan Mod)