Duct9143 I think that's up to your threat model (what do you want to protect and from whom). Also consider this:
- you can pay cash
- you can pay with gift cards
- you can use debit cards that you replace regularly
- you can pay with virtual or single use credit cards online (check privacy.com or Revolution depending on your location)
Since you mainly ask for privacy implications, paying cash would be best for privacy. If you require cc payment, probably using multiple debit cards from different institutions would help with privacy (compmentalize your data among multiple companies).
If the decision was only between Google Pay vs a singular cc, I'd probably still use the cc since my purchases will give Google too many identifiers, making my throwaway account much more personal. But that's just my 2 cents, both solutions are not very private in the first place...