Basically, yes it does.

Everybody already gets 5x GP points per $ spent on eligible spend at Hyatt. (so that excludes things like parking and taxes I believe). Every member of Gold Passport gets that. (and if you are Diamond, you get a 30% points bonus on your eligible spend; I think 15% bonus if you are plat.)

The credit card gives you another 3x per $ on total spend at Hyatt for a total of 8x on eligible spend plus some extra for other spend charged to a Hyatt on the Hyatt credit card because it's coded as Hyatt spend.

But you would still get that 5x on eligible hyatt charges on any stay and you could also charge your stay and get 3x Citi Thank You points when charged to Citi Prestige MC. (Just as an example.)

So you aren't getting that first 5x on your eligible spend on your folio because you used the Hyatt credit card. It's not exclusive to that card. You are getting it as a member of the Gold Passport program.

-David