Snippet of the Benefit Terms (with my emphasis added) below...
You can earn up to $179 in statement credits per calendar year when you use your Platinum Card to pay for your annual CLEAR membership.
That suggests to me that it will only work on charges related to your membership but honestly CLEAR Pay is not explicitly excluded by the ToS here so I think that leaves it hanging as a maybe? Looks like this is a situation of "Be the datapoint you want to see in the world!"
Edit: I am not seeing information anywhere that suggests you can use CLEAR as an alternative payment method, but honestly the CLEAR website is a little sparse/vague on details. Where are you pulling that quote from?
Double Edit: The only information I can find on Pay with CLEAR is a now-deleted TPG article that references the ability to buy Beer/Concessions at SafeCo/CenturyLink Stadiums in Washington.
https://web.archive.org/web/20191023...r-fingerprint/