Good to know. We don't have burner phones over here, and can't verify, however I thought their SIM cards were locked so that they couldn't be used on other devices.
Where I live (NL) you don't even need to buy a whole phone, anonymous prepaid sim cards don't require any ID to purchase.
The burner phone would be a better option IMO. Phones can be traced with their IMEI so if before or after using it with an anonymous SIM it's/was used with a personal SIM, it will be easy to associate the phone owner's name with the anonymous SIM.