Small diesels could be an option but they're harder to pull start for a given size.
I once needed to jump-start a small marine diesel, many miles from land...
There was a small lever that cuts compression. You have to get it spinning really fast before restoring compression! It's definitely a lot of work!
EDIT - Here is a cheap modern small marine diesel [1]. The operation manual suggests that you don't have to do anything to get it spinning quickly, you just have to crank it 10 times, put away the crank handle, and then flip the compression switch. That's progress!
[1] https://www.yanmar.com/marine/product/engines/1gm10-marine-d...
Cranks and decompression levers are gone for at least 30-40 years now tho.
They're my kryptonite, but I accept it's mostly my ignorance.
Air compressors have more valves and gaskets that are vulnerable to oxidation, especially in salty environments, so I'd have thought the upkeep between the two, the two stroke would be easier.
Having good, fresh fuel on an oil rig. They need an engine that can run on crude.
Maybe there are other concerns for an oil rig.
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"
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.