The very first card that Privacy.com recommended I replace was the card I have for paying my Spectrum bill, which Spectrum is obviously already accepting. Privacy.com even linked me straight to Spectrum's billing page to replace the old card with the new "improved acceptance" card. Imagine my surprise when Spectrum rejects the new card with "Prepaid and gift cards are ineligible"
EDIT: I regenerated the card again and Spectrum accepted it no problem. No idea what happened there. Maybe there was something funky with that particular card number, or maybe I mistyped the previous number? Either way, it's working now and I've had no problem with the rest of my card replacements so far (though it has certainly been a bit of a slog).