Semoochie, I haven't seen the PBS show in question. You're quoting your mom second hand. I can't find any reference to the specific interview anywhere online.
The only thing online mentioning "Three Dog Night" and "mafia", "organized crime" or "mob" is a Quora post from two years ago---in a thread about Tommy James' book which asks the question "Who in the music industry have crossed paths with the Mafia?"
It's one person replying "Supposedly 70s band Three Dog Night pissed off the Mafia so hard they could no longer get hits through radio play and such." No up or down votes, no replies, no follow-up.
Online interviews with Danny Hutton say nothing about mafia interference. Cory Wells addresses the band's decline in a 2008 interview with Las Vegas Weekly this way:
"We were the darlings of rock there for awhile. We were the darlings of Rolling Stone magazine, until we got too successful. They loved us until we had success, then we weren’t cool enough or edgy enough for them (laughs)." (Full interview here:
Lively banter with Three Dog Night co-founder Cory Wells)
And Chuck Negron (no longer with the band) told this to the San Jose Mercury News in 2012:
“The only downside for me, of my career in Three Dog Night, was that the egos never really settled in, like most things do. Can you see Scotty Pippin going, ‘Listen, this Jordan guy’s got to stop shootin’ so much!’ I’m the star!’? No one ever got comfortable with the fact that the first million-seller was me singing ‘One,’ and then the record company, being the record company, wanted another single with my voice. And I happened to have another one in the can that was a hit, ‘Easy To Be Hard.’ So you know record companies — if they’ve got something that’s not broken, they don’t want to fix it.... You bring in your song and you do it. And the other guys are hoping it dies,” Negron says with a chuckle.
...At any rate, pressure got us. We were making so much money that the business end of the people just wanted us to work. We’re doing two albums a year, touring 210 days a year. We’re going from guys that look in our 20s to guys that look in our 40s. And we were having sex with anything that has a pulse,” Negron says, laughing.
The band’s implosion concluded in 1976. “Unfortunately, the business, chewed us up and spit us out. Because we let it. And then there were the drugs.” (full interview here:
https://www.mercurynews.com/2012/08/15/the-dark-one-dog-night-of-chuck-negron/)
To the question of "Did Three Dog Night lie to PBS?" Nobody does media interviews under oath. It could be something whichever member of the band was being interviewed believed, or believed at that time. All I can tell you (apart from all of the above) is that, as big records from the Eagles and Fleetwood Mac were landing on my desk in '75 and '76, I was still getting calls from ABC Records trying to get me to add the latest Three Dog Night.
And yes, you're right. Moe died in 1975. Still, Tomas Estefan and his male classmates might have preferred the Three Stooges in their then-current state to Bobby Sherman.