Why weren't actual NFL teams used?

Because the film is not a very illustrious view of NFL players or the NFL in general. Eric Hamburg, co-producer for Any Given Sunday:

Any Given Sunday was delayed four times from its initial April 1998 start date with negotiations between Stone and the NFL also disturbing production. “Early in the process, I was in touch with a woman at NFL Films and she sent us all kinds of material—film, video—and they were eager to cooperate,” Hamburg says. “But then they got wind of what the film was going to be.”

Jamie Williams, a former tight end for the 49ers, and technical consultant for the film:

“The NFL didn’t want people to see that these football players are just like the rest of society,” says Williams, who played 12 years in the league. “These are guys that do too many drugs or spend their money unwisely, but then they go out on Sunday and are gladiators. None of these guys are boy scouts, not even the quarterbacks.” The NFL declined to participate, reportedly issuing a memo to teams suggesting they not cooperate with the film's producers; Concussion uses NFL footage, logos and images without the league’s permission.

Concussion being a 2015 film revolving around football, which itself had many issues with the NFL treatment (Read: Lots of CYA not to piss the NFL off).

Like many other organizations, (cough cough NASA), the NFL is very concern on how the NFL or its players are depicted. Something saying NFL coaches and doctors and players are all felonious bastards would be a big no no.

If there is a real villain in Any Given Sunday, it’s the culture of football, with James Woods’s Harvey Mandrake, the crooked physician, and Christian Pagniacci, the team owner played by Cameron Diaz, representing products of that system. Pagniacci instructs Mandrake to fake test results that would clear Shark Lavay, the heart of the defense, to play. Later on, she says of Lavay, “No one is going to sign a $2 million concussion case…Cut him, sign him for 30% if we need him.” There are no guaranteed contracts in the NFL.

Even Al Pacino’s coach Tony D’Amato is complicit. “You play hurt,” he tells Cap Rooney. “You just need the needle.” He’s also in the room when Shark Lavay signs a waiver exempting the team from responsibility in the case of paralysis or death. “I think a lot of those things weren’t realistic,” says Barry Switzer, one of three coaches to win both a Super Bowl and a college football national championship. “No coach ever makes a player go on the field when he has a chance to hurt himself. Never.” Switzer, who cameos in the film as an announcer, however, finds Pacino’s “Game of Inches” halftime accurate. “Oh yeah, that’s coach speak. We’ve always known it’s a game of inches.”