Answer
Yes, airlines check the passenger's right of entry before boarding, because if the passenger is denied entry at the border, the carrier may be obliged to take them back at its own expense. Therefore, questions at the check-in desk are not a formality.
The carrier may request a visa or proof of eligibility for visa-free entry, a valid passport, a return or onward ticket, an address of accommodation, and confirmation of the itinerary. The requirements of transit countries are also important: they can be stricter than the rules of the country of arrival.
If online check-in fails, it does not always mean a final refusal. You should contact the check-in desk and a senior specialist, with your documents at hand. However, responsibility for preparing the passport, ticket, transit arrangements, and right of entry still rests with the passenger.