How did chants and songs evolve in soccer/European sports and not American sports?

It IS a big part of the American Sport experience. Perhaps most of all in College Football.

For all the same reasons, build the team identity, link it to the fans, and show support.

But then with the rise of college bands something shifted, they began writing songs too, so you have the almost unique instance of school not just having an "alma Mater" but now having Fight Sings and Victory marches that their bands would play and fans would sing at events in addition to normal chants. Obviously most schools and teams across the world have some sort of anthem or song, but started was taken to heart most of all in the ranks of US collegiate athletics.

The earliest recognized one is Boston College's "For Boston" in 1885. https://www.youtube.com/watch?v=R3ckYY9Vtrs

And many are quite famous like Notre Dame's https://www.youtube.com/watch?v=o_NxSTX0jkc

And some include synchronized crowd movements like Texas A&M's: https://www.youtube.com/watch?v=t3N4ajREoHw

Though that did not stop other schools from also just developing short cheers that could be done with or without the bands, such as the University of Alabama's "Rammer Jammer" :https://www.youtube.com/watch?v=cCUkc26kQbI

Or Ole Miss's Hotty Toddy: https://www.youtube.com/watch?v=dRNeSESId30 performed to the same tune as Alabama's.

Or just playing a song so often it is used as such, like Tennessee and "Rocky Top" https://www.youtube.com/watch?v=KsIJ3L7Sj3c

Or in some cases cheers can just be like everywhere else a call and refrain across the stadium like my Alma Mater, Virginia Tech: https://www.youtube.com/watch?v=KfJQGf0nelg

Or many schools have turned to more contemporary music as well, Wisconsin's Jump Around for instance: https://www.youtube.com/watch?v=tODlPxhbGF8

Or Virginia Tech again with our entrance song "Metallica's Enter Sandman" being used in key moments to amp the crowd too: https://www.youtube.com/watch?v=blzftASduNc

And because College Athletics predated Professional sports they took their ques from the college game. So you have the Washington Redskins fight song: https://www.youtube.com/watch?v=kPVfFzbRuc4

or the J-E-T-S JETS JETS JETS cheer for the New York Jets as examples.

So I would say simply that your question is based on a false premise based on a limited exposure to the American sports landscape.

For football too you used to have a rule where if the visitors went to start a play and they could not hear to communicate the Officials could penalize the home team. That ended in the late 60's so it was really only after then that the use of "home-field advantage" took off.