OK, this is probably the oldest and most common point of confusion due to terminology between mechanic and car owners. (Not to mention a common source of easy $$) In this case you may be on the right track, but I learned long ago to make sure, so please bear with me.
The key is to distinguish between “cranking” (or cranking over, or turning over), and “starting” (or firing, or catching). Cranking is what happens when you first turn the key and hear the engine spin before it starts. This is often just a blink of an eye because cars start easily these days, but it is that time before you release the key and it goes back to the run position. Starting is when the engine starts to run on it’s own, at which time you let go of the key to stop cranking. (or when you pull the hand crank out of the front of the crankshaft on your Model A.
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Part of what you describe sounds like it cranks OK but won’t fire, but sometimes fires but then dies. If that’s true, then it is not the starter or the battery*. If it cranks very slowly then it may be either the starter or battery, but more likely the battery. (It doesn’t need to crank very fast to start.) But that doesn’t explain the start and die scenario. If you have enough juice to crank and start, you should have enough to run for at least a bit even if the charging system has failed.
If it cranks slowly or not at all, start by pulling and cleaning the battery terminal clamps. If it cranks like mad but won’t fire, then it’s likely fuel or spark.
* I suppose a bad connection might allow you to crank and start, then the vibration of running shakes the connection loose and it dies. There are lot’s of remote possibilities like that, but we’ll stick to the probable.
