You don't always have to provide a phone number, either. Most of the time yes, but I don't think the phone number is actually required to complete a credit card transaction.
There also used to be credit providers - maybe there still are? - that would let you generate a new card number for each purchase. That too may accomplish nothing from a tracking perspective, since the credit provider would be able to match up all the numbers. But, if I had that capability, I would use it.