That's not what the webpage says, which states "An infant for whom a seat is purchased must be properly secured in an approved child restraint device."

The way I read that, if you buy a seat for an infant, you must use a child restraint device. Not, as you suggest, use of a child restraint device requires that a seat is purchased.

In practice, YMMV with AC FAs. If the OP selects their seats strategically and an adjacent vacant seat is available as a result, some FAs will let you use the car seat. If not, you can gate check it for free, so no harm no foul. Just be straightforward and ask the GA right before boarding if Seat XX is empty or not...