My limited experience with provider bills is that even two months “late”, none have ever referred the bill to collections. Nor did they ever charge the “late” fees that their bills had threatened. I’ve never come across a provider that demanded a credit card on file – and chargebacks would ensue if they did run the card without notifying me – but my experience certainly won’t be a consistent across the country.

I’m not sure what you mean about the part involving tax evasion, but broadly speaking: if a card is going to be held on file – whether for a rental car or something else – debit cards are not advisable. Also, my cursory understanding is that HSA cards should only be used for point of sale transactions, since apparently it could sometimes be declined for card-number-entered transactions.