Well, I think the devs mentioned that one of their goals post release would be to flesh out the star base system - either via expansion or dlc.
I agree military bases just don't feel useful right now.
although it would be a huge change, I think you shouldn't be able to stack economic star bases on a single planet. It takes too long, results in massive micro to min/max, and is generally poor game design. It would be a big change, but I'd like to see only 1 of each star base type per planet. ie, economic star base boosts the planets raw stats, maybe hosts some off world pop, etc. mining star base mines nearby asteroids and resources. influence star base projects and accelerates culture. military star base boosts nearby fleets, repairs, and helps defend against invasions. etc. I agree they should be sponsored by shipyards rather than constructor spam.
as for military star bases, a really simple change to boost their usefulness would be to have a very low tech module that rapidly repairs ships - in say 2-3 turns. that alone would make them really useful in early game until repair tech tree becomes available in mid game. also a module that either heavily amplifies a planets invasion defense, or allows for fighters to "intercept" unescorted transports.