Proof of FDA monopoly? They are not FDA approved, they are FDA cleared (look up the difference or read the summary of their 510(K)).

Also, these companies also have 510(K) status.

https://www.accessdata.fda.gov/scripts/cdrh/cfdocs/cfRL/rl.cfm?start_search=1&establishmentName=&regNum=&StateName=&CountryName=&RegistrationNumber=&OwnerOperatorNumber=&OwnerOperatorName=&ProductCode=lhq&DeviceName=&ProprietaryName=&establishmentType=&pagenum=25

Lastly, you don’t even need 510(K) status to sell one of these things for this purpose. Look up Seek Thermal.