> You'd rather hand over your credit card than your phone number?
You know, that was my first reaction, too. But really, my phone number is much more integral to my identity. I can cancel or change a credit card number pretty trivially and then it's useless to you.
Many credit card companies make it easy to generate one-off card numbers/“virtual cards” you can use to subscribe to services that are hard to cancel or otherwise questionable (so you can cancel just the card you used for that company).