Square/cube law strikes again. You can only run an automotive starter continuously like a minute or so (at absolute best) before something melts down and the system is scrap. Scaling the power cube up by a factor of a thousand or so and the cooling square surface area up by a hundred or less and suddenly you get like one revolution of the motor before the starter windings and relay/contactor melt down. Not pretty. Of course you could make everything unimaginably immense but that starts making air tanks and air compressors look cheap.
I think it was because the system was simpler and more reliable with no moving parts besides valves, just a giant air bottle with pipes leading to each of the cylinders. No motor to worry about or lead-acid batteries to maintain. The whole system probably took up way more room than a starter motor would but excessive space or weight is not a big concern on an aircraft carrier.
I think it also had something to do with age - newer vessels (submarines, at least) use a large lead acid battery array for emergency power when needed... and by emergency power, I mean enough power to start the emergency diesel generator.
Submarines do have a large lead-acid battery for emergency power. In fact, the WWII-era boats had bigger and better batteries.
However the battery isn't there to start the Emergency Diesel Generator. Submarine EDGs are air started, for many very good reasons.
We still need the batteries since it's possible to suffer a loss of all AC power while the boat is submerged far beyond feasible snorkel depth, and have to recover from it without broaching.
I don't know much about this sort of thing, so I'll ask the obvious dumb question: why no batteries to start the diesel generator? Was it too big?
Edit: Reading on, it seems like compressed air is the norm. Is it more space efficient than a battery big enough to start a large generator?