“Do they hold the rights to the songs?”
Odds are no.
The way music publishing worked in Issac Hayes’ day:
1. The song’s author would sell 100% of the rights to the song to a music publisher.
2. In return, the author would get a percentage of all publishing income from the song (usually 50%) in perpetuity.
It was a terrible system, designed to rip off the songwriters, but it was the system.
It’s the reason in the late 60’s large artists (The Beatles, Led Zepplin, and such) formed their own publishing companies. So they wouldn’t have to give away their rights (and 50% of their income) to a third party.
Unless Issac Hayes has purchased back the rights to his song catalog (which many artists have done) odds are he doesn’t own the right and they are administered through a 3rd party.
Unless that 3rd party (who is the actual owner of the rights) objects, Trump is in the clear.
Trump has been through this rodeo before with artists refusing to allow him to use their songs. It’s likely they licensed the song through the normal processes.
Don’t the rights to use recordings become public domain after 50 years?