I can’t envision how an oil leak would damage the starter, but I’ll take your word for it. An oil leak could damage an O2 sensor though. But that wouldn’t usually throw a CKP code.
hmmm … well the CKP sensor is a gadget that produces a pulse used by the engine computer to synchronize the ignition system firing pulses to the crankshaft position. So that the spark occurs at the right time in relation to the piston position, a little prior to the top of the compression stroke. So if it wasn’t providing that position accurately, it could indeed make the engine hard to start and run erratically. Loss of power, pinging, stalling, etc. Do you have any of that?
It’s unusual that a car would start if the CKP was dead, but I suppose it is possible your engine design has a back up method which allows the car to still run – albeit not as well – even with a failed CKP. Or the CKP is working, but not working like it should.
I think what I’d do in this situation is remove the current CKP, that’s the new one, right? And take a look at it and whatever it senses, usually the edge of the flywheel. Everything new looking and free of oil and grime? If not, clean it all and make it look like new. Rotate the engine by hand for a visual check to make sure the entire flywheel edge is clean, and the the flywheel isn’t getting damaged or chipped by the new starter motor’s gear. That could cause this symptom. And visually verify that the connector the CKP plugs into isn’t corroded or coated with oil and grime.