(ABC News)

A new season starts on Thursday for a National Football League that's either astoundingly prosperous or in crisis mode.

Amid record revenues and fan interest, the NFL is facing harsh scrutiny on its handling of sexual assaults, head injuries, substance abuse, and even the nickname assigned to one of its most prominent franchises. The issues have attracted widespread attention on Capitol Hill, casting a shadow on the start of the season.

WATCH: NFL Cracks Down on Domestic Violence

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WATCH: Washington Redskins Name Under Fire

In the latest episode of "Capital Games," Sen. Marco Rubio said "the league is going to struggle to adjust" to a media environment that highlights misbehavior that would have merited only local headlines - or stayed hidden altogether - in a previous age.

Listen to the full podcast HERE .

As for the Washington Redskins nickname - a name President Obama, more than 50 Democratic senators, and a range of Native American groups have deemed offensive - Rubio said it isn't for public officials to dictate.

"I'm not a game-show contestant; I'm a US senator," Rubio, R-Florida., told me and ESPN's Andy Katz in the podcast.

Still, Rubio said, the growing concern about the Washington franchise's team name "shouldn't be ignored."

"The decision is up to the owner [Dan Snyder] to make," Rubio said. "I think he should take into account that there are people that are offended by it. I know there are others who feel strongly about what the name means and don't see offense to it. He's in the PR business; he has to sell tickets. And ultimately he's going to face accountability from a business perspective for whatever choice he makes."

Rubio stopped short of echoing calls of many of his Democratic colleagues to change the nickname. He said Snyder "should listen to voices" objecting to it, "but in no way should he be forced" to change it.

A new poll released this week found that nearly a quarter of Americans now believe the name should be changed. That means seven in 10 are fine with keeping it, but the number supporting a change has grown dramatically since pollsters first asked voters about it more than two decades ago.

More broadly, Rubio said the league needs to do more to teach players that they are "role models," not just athletes. Just last week, NFL Commissioner Roger Goodell announced a new suspension policy for domestic violence offenders, after harsh criticism of the light punishment issued to Baltimore Ravens running back Ray Rice.

"I think both the league and the teams should take more seriously how you discipline athletes who, in fact, are proven guilty," Rubio said. "Now obviously there is due process, and an accusation alone shouldn't lead to that, but if it is proven that an athlete has conducted something that is unacceptable such as violence against a women or a sexual assault, they should be suspended severely."

With congressional scrutiny increasing, Rubio cautioned that members of Congress should generally be focused on more important things than oversight of professional sports.

"There's some limited role, but I don't think it should be the center of our work" in Congress, he said. "There is a role to play, but a limited one."

Also on the podcast, ESPN NFL business analyst Andrew Brandt talked about the important decisions ahead for Goodell on a range of issues that have attracted public attention. As for the Redskins name, noise and pressure from politicians is unlikely to influence team ownership as much as sponsors' pressure, Brandt said.

And Rubio, who is considering a 2016 presidential candidacy, promised to keep up a tradition started with President Obama and Andy Katz: He said he'd fill out NCAA basketball brackets if he's in the White House.

"I don't know what the chances are, but if that happens, not only will I do the bracket, but mine will actually be successful. I'll actually get them right," Rubio said. "I've never won a bracket before, so that's how I know I'm due."

"Capital Games with Andy Katz and Rick Klein" is a part of the podcast series ESPN Perspectives, with original audio programming on issues across the sports world. The program explores the intersection of sports and politics, through interviews and analysis, and can be downloaded free via iTunes or on the ESPN Website.

Listen to the full podcast HERE .