> You could get by with requiring a unique phone number
In the US, anyway, you can also get burner phones for about $10 at local stores. I do this routinely if someone is requiring a phone number to register for something that I really want to register for.
It's no more or less disconnected than any other store purchase. You can assume there are cameras in any store. You can indeed pay cash. The only records generated are the usual sales records, and if you're concerned about minimizing those, then you use the same mitigations that you'd use with any other purchase.
Personally, I'm not concerned with that level of anonymity, though. I just don't want to give my actual phone number to random companies.
In the US, anyway, you can also get burner phones for about $10 at local stores. I do this routinely if someone is requiring a phone number to register for something that I really want to register for.