I think the key here is the fact that you have paid the room rate to the 3rd party agency, who in turn paid the hotel for your stay (check your folio to see what was used for the payment - personal credit card or maybe a company account). If that is the case then no wonder you have not received the points. The payee and the GP member names must match for the stay to be eligible for stay credit/points. Otherwise I could be added as a guest to a lot of reservations of my friends/family members who infrequently stay in Hyatt hotels or who don't want to bother with the FF programs (blasphemy, I know) and reap the rewards w/o spending a cent.
This rule is outlined in the T&Cs (found
here) under the section "Earning Hyatt Gold Passport Points as a Registered Guest"
3. In order for a Hyatt Gold Passport member to receive Hyatt Gold Passport points, the bill for charges must be in that member's name and member must personally pay for all charges.
You argued that nothing in the T&Cs requires you to pay Hyatt directly but that you paid personally for all charges. As far as Hyatt is concerned, they go by who paid them as they really don't have a way of determining whether you personally paid the 3rd party agency or not.