This usually is the least of the airline's concerns. If you're flying on a round-trip ticket, then the airline simply waives any change fees or other requirements and flies you back to your point of origin at your expense - at that point, the round-trip ticket is considered "used" in both directions, as the airline actually transported the passenger as ticketed in both directions.

The real issue is the potential for fines by the country that refuses entry of a passenger. If the US refuses entry to an inbound passenger and ICE determines that it's something the carrier should have caught prior to boarding the inbound passenger, they can fine the airline up to $10,000 per occurrence. Many other countries have fines of similar size. That is, immigration authorities in most countries want to provide a strong incentive for carriers to catch paperwork problems prior to the passenger being transported.

Even in the US with its strong union environment at many carriers, it's not too difficult for a gate agent to get fired for boarding a passenger with incorrect documentation for the destination if the airline gets stuck with a big fine. This is why gate agents and supervisors at airports seem to take this issue personally - it is personal to them, as their job may be on the line.

Signing a waiver may help in some situations, but in the case of a refusal of entry, the rejecting country is going to collect the fine from the airline, and then the airline has to come after the passenger civilly to collect off of the waiver. They may or may not want to deal with it.

As others have said, Timatic is generally considered an authoritative source by airlines, so it is a bit odd that Timatic would say "Nationals of USA can enter on passports or passport replacing documents valid for the period of intended stay" yet have this questioned by the check-in staff. Especially since both the Thai embassy and the US State Dept's websites mention the 6-month rule, perhaps the change in the rule is recent. I could see how this could cause confusion. Hmmm... I'm not sure what I'd expect an airline to do in the case of a recent rule change that is listed only in Timatic, but not on the actual country's website, the US gov't website, or known by people at the country's flag carrier. Although generally speaking Timatic is considered authoritative, I can see how the actual country's website could be considered more authoritative than Timatic, and unfortunately in this case, if it's out of date, the passenger may suffer.