Hyatt has an official policy stating that FHR is elite qualifying. There is no such policy for HC. Similarly, many Marriott stays (IME) do post to my Bonvoy account when booking through FHR. I do not have any experiences with booking Marriotts through HC but the lurkers have (recently and in the past) confirmed FHR elite eligibility without confirming HC elite eligibility.

Based off of these 2 factors, I believe HC is NOT elite qualifying for Hyatt and Marriott. I would be curious to see if there are any data points for this with IHG as there seems to be a few Kimptons in the HC program.