There are other claimants but the Stooges are the strongest because of their direct influence on the Ramones who would become the classic sound of American Punk. Some of the Ramones saw the Stooges perform, their manager Danny Fields was an ex Stooges manager but more crucially the sound of their debut is not a million miles away from the third Stooges album Raw Power or the faster tracks on their 1969 debut or 1970’s Funhouse. It’s worth noting that other bands that were noteworthy proto punk/glam acts the Flamin’ Groovies, Alice Cooper & the New York Dolls were also influenced by the Stooges, some even changed their style after seeing them.
The Stooges fellow band, the MC5 were also protopunk and their political message would influence more politically minded punk acts in the 1980s. The Stooges were also covered by the Sex Pistols and the Damned, early UK punk groups that along with the Ramones would influence several early US Punk acts including Black Flag, The Dead Kennedys, The Germs, Bad Brains, the Dead Boys and others. So the biggest precursor to American punk music, as we define punk nowadays, is the Stooges. The band that was most influential overall on 70s Punk/Art Punk & Proto punk acts at the time was the Velvet Underground. We just don’t really call those arty Bohemian punks that followed their example “punk” anymore, the Ramones have become the punk yardstick.