Private hospitals usualy do the job with the immigration on your behalf, it is part of their public relations service. (not only for covid related cases.)
Home quarantine only if you come from red flagged provinces. Doesn't apply if you go straight from ASQ to airport. A friend of mine arrived in Hat Yai last week with no issue.
One thing I don't understand. How come the immigration accept an insurance with a high deductible amount? Isn't the insurance supposed to pay health expenses from 1 to 400,000 baht?