That doesn't work when there are travel restrictions based on the passport being used to enter Thailand. The airline is responsible for ensuring that the passenger meets the requirements for entry based on the passport presented to them for check-in. Please see my detailed response above explaining this. This situation has been thoroughly discussed in all the dual-national travel groups for how to handle travel worldwide during the pandemic.The Thai passport needs to be used for check-in as the US does not have exit immigration procedures requiring the US passport to be shown. Otherwise the airline has to do a complicated override system that has lead to many missed flights and drama at check-in to handle the fact that the name and passport being used for check-in is not the one that is being used for entry at the destination. When there are no differences in the entry restrictions between both passports, that is the only time it doesn't matter which passport is shown for check-in in the USA vs the one used to enter the destination.
I think your question is not about what passport to use when entering Thailand as others are answering, but rather what passport to use for the airline ticket and at check-in. This is covered a lot in the various travel forums regarding dual-nationals traveling around the world. The US does not have exit immigration clearance, so you should buy the ticket and check-in using her Thai passport. Otherwise, using her US passport can create difficulty at check-in because if the airline enters her US passport, it will alert them that they must follow all the entry requirements for US nationals to enter Thailand via visa exempt, and she may not meet all the requirements (return ticket, insurance, etc.). Technically, if she used her US passport name, they can override with a notation that she has a Thai passport as well but the problem in this covid era is that airlines are super strict about their personnel following exactly the entry documentation requirements listed for whatever passport is shown in the system, which in this case is US, so it will likely cause significant delays involving calling supervisors and arguing with them, so why do it? She can continue to show her Thai passport to get through TSA and at the gate, since as I said, the US does not have exit immigration so all they care is that the passport name matches the name on the ticket. In the extremely rare case that CBP officers do a spot-check at the gate for illegals, she just needs to show the officer both passports and they will notate it in their system.
Did you get an insurance document from AIG that looks like this? (this was the policy format for last year/earlier this year so it may have changed but should be this type of similar policy doc.) If that is the case, many travellers had posted before that the consulates would miss the line that specifically mentions COVID coverage and would deny the COE until the same page was resubmitted with the COVID coverage line highlighted, and then it was always approved. In my example, it is the last line where I drew the red arrow. If this line is in your policy doc, then resubmit it with that line circled or highlighted. This is assuming your coverage term isn't the issue, as the AIG coverage is for the exact dates of your roundtrip Emirates ticket. If you do NOT have the policy doc, then you need to email
********************
with a copy of your Emirates roundtrip e-ticket and ask them to send it to you.
I'd be cautious. Usually signing on a plain piece of paper is used so someone can duplicate your signature either as a scan or to make a signature chop to stamp it on documents or cheques. If they are submitting any applications for you at the embassy/consulate, I don't see why they can't send you the documents for your signature to scan back to them instead. Even without a signature line, it is normal in Thailand for you to have to sign just anywhere on the bottom of every page of a contract or application document, but again, you should be signing the actual pages of the document and not let them duplicate your signature onto every page via a scan or stamp of it because that would allow them to use it for anything for eternity.
my friend with Hong Kong passport was just rejected for his first Covid extension at Samui immigration because they said there are plenty of flights between Bangkok and HK so he doesn't qualify. He is currently on his 30 day extension on a visa exempt entry.