I have a French passport and a US passport. If I book a flight with one passport and the airline employee at the check-in desk refuses it due to its physical condition, can I check in with the second passport? Assume I don't have to present any visa to the destination country's immigration officers that can't be done on arrival to be allowed to enter the country.
Typically, yes. The airline doesn't really care what nationality you are or which passport you use, all they need to know is that you will be accepted at the destination country, and they'll accept whatever documentation you have for this.
The one major caveat to this is countries that insist on passenger info being registered well in advance, notably the US. Then again, the US also requires that everybody have a valid ESTA, so swapping passports is likely not an option anyway.
Source: I've done an unplanned passport swap at check in a few times. Details:
- Going to Vietnam (visa needed at the time), check-in says my passport is expiring in less than 6 months and thus not valid for travel. I show them my second passport with plenty of expiry (but no Vietnamese visa!), they accept this instead and all is good.
- Returning from the US, my son is refused check-in because his passport's last name does not match the booking last name. I show them his second passport where the last name does match, after some grumbling and checking with a supervisor it's accepted.
But your mileage may vary, in both cases there was a distinct possibility that the agent would have said no.
Your question seems to depend on the common assumption that airline bookings are bound to a specific passport. That assumption is entirely incorrect. One man even took a woman on a round-the-world trip simply because she had the same name as his ex-girlfriend.
I routinely use different passports for different flights on the same booking (a US passport and an EU passport) because I am typically flying between the US and the EU. Once ETIAS goes into operation, this will likely become even more common, as there will be additional motivation for dual US/EU citizens not to use their US passports to fly to the EU.