The main carriers offer prepaid plans, and there is no postpaid plan that doesn’t throttle speeds after you go over a certain amount when the towers a busy.
Most people get suckered into signing a contract and using a “postpaid” plan, where you get the service for a month and then pay for it. That requires a credit check and credit reporting, since you get the service before payment. You don’t have to give out your SSN if you sign up for “prepaid” cell phone plans, which offer less discounts and benefits but are generally cheaper for the service they provide. The only catch is you pay for the month before you use it, but this makes canceling as easy as stopping payment.
If you want to use Privacy, you can reprogram a magnetic stripe to work with privacy cards, you’ll just need a new card for each merchant.