> A totally anonymous phone card would be a bit tougher
Depends on where you buy them. The internet is big and you can have stuff delivered anywhere, including somewhere that doesn't have your name or address.
I think a common form of theft is for someone to be defrauded by an order sent to a third party, as if it were a gift and for the package to be picked up from the third party similar to porch pirate activities. I saw a short documentary about this performed at an semi-industrial scale where the pickup was done by a car service person as a side-gig.
The US seems ok but your mileage may vary by domicile:
As of early 2021, 157 governments required some form of proof of identity before a person could purchase a SIM card, but what form of ID and what other information may be required varies.