Batteries are great when they have charge. What happens if the generator doesn't want to start the first, second, and third time? How many start attempts do you get before the batteries are dead?
The hand-pumped air compressor is the tool of last resort. You can try an engine start if there's someone there who's able to pump it. You don't have to worry about how much charge is left in your batteries or whether or not the gasoline for the 2-stroke pump engine has gone stale. It's the tool that you use as an alternative to "well, the batteries are dead too, guess we're not going to start the engine tonight... let's call the helicopters and abandon ship"
The data center where I work has large diesel generators for power cuts. They are electric (battery) start. There is no capability to start them manually. The batteries are on maintenance chargers that keep them in good condition. The generators are started and tested every two weeks.
Could the batteries be dead and the generators not start? I guess but it's very unlikely. I get that on an oil rig it might be a matter of life and death and you need some kind of manual way to bootstrap but there's not much that's more reliable than a 12V lead-acid battery and a diesel engine in good condition.
Also, the data center is probably in a city, surrounded by infrastructure that could be used if necessary. An oil rig is in the middle of an ocean, and has to rely on itself.
Lead acid batteries are not exactly what I would call reliable. They require a lot of constant maintenance to ensure that they will work when you need them and they can easily degrade in such a way that they maintain voltage and appear to be good but then fail to deliver the needed amps when you demand them. This is made much worse in cold weather. Finally, if allowed to freeze when they are moderately drained, then the accumulated water inside will freeze and drastically shorten their life span.
I think I'd take Lithium Ion batteries over lead acid for almost every conceivable use-case. They are superior in almost every way. Lighter, less likely to leak acid everywhere, better long term storage (due to a low self-discharge) and better cold weather discharge performance. The only drawback would be a slightly increased risk of fire with Lithium.
I worked with a telecom provider's data center that ended up having a quad redundant diesel generator failure during the first cold snap that took the Texas grid offline a few years back. They had at three fuel supplies gel and then failed to start. The fourth, as I remember, just didn't try to fire.
It's unrealistic, and if one power station is unable to use their batteries to start their emergency generator (through the absurd incompetence you describe, or more likely through a major fire, flood or assault) the grid can be started from a different one.
The hand-pumped air compressor is the tool of last resort. You can try an engine start if there's someone there who's able to pump it. You don't have to worry about how much charge is left in your batteries or whether or not the gasoline for the 2-stroke pump engine has gone stale. It's the tool that you use as an alternative to "well, the batteries are dead too, guess we're not going to start the engine tonight... let's call the helicopters and abandon ship"