Many payment providers would want websites to implement CAPTCHA for blocking spam and fraud attempts.
I'm sorry, but you're just going to have to walk to a store and pay in cash if you don't want to have any data tracking done at all. Online you'll often have to pick one or the other: data tracking, or flimsy security/data protection. The phone solution is appropriate in my humble opinion, but you're welcome to hold your own views on your principles. If you're hard set against a company tracking ANYBODY via 3rd parties to that level, then I bet you will be very hard-pressed to ever find a computer through an online marketplace from ANY company that will fit that bill perfectly and suits your other needs.
The same government that presented another deal so good you couldn't refuse!