Gebbia, a redshirt freshman from Calabasas, California, lost an offseason-long battle for the starting job to true freshman Adrian Martinez over the weekend.
Gebbia skipped practice Monday and withdrew from the university. An ESPN 300 signee in 2017, Gebbia threw for the second-most yards in California prep history.
"We want guys who want to be here," Frost said.
The Cornhuskers are left dangerously thin at the position in Frost's first season. Walk-on sophomore Andrew Bunch has fared well in practice, according to coaches, and freshman Matt Masker joined the team in August as a walk-on.
Frost said Tuesday that Nebraska will renew its effort to seek a waiver from the NCAA for Vedral's immediate eligibility.
"In the big picture, I hope that college football doesn't become transfer mania," Frost said. "It's hard to manage as a coach. You do what you can for kids, and I understand their perspective. They want to play. It would be hypocritical of me to talk about how much we care about the players, then not allow someone to do what they want to do or want what's best for them.
"The flip side of that is we preach to our guys all the time, team before me. And I think it would be hypocritical of us to have someone on our team that's me before team. Sometimes those things are going to work out the way that this just worked out. We've got to do our best as coaches to manage the situation."
Gebbia was recruited by former coach Mike Riley along with teammate Keyshawn Johnson Jr., who left the Huskers before last season. Gebbia redshirted as Nebraska endured a 4-8 finish and looked to fit in well with Frost, who brought his up-tempo offense from Orlando.
Gebbia and Martinez, the headliner in Frost's first recruiting class last December, battled through the spring and into August. Nebraska announced its QB pecking order on Sunday with the release of its depth chart ahead of the opener against Akron this week.
Martinez said he learned the news Saturday from quarterbacks coach Mario Verduzco.
"It was close," Frost said Monday. "We picked the guy that we felt would give us the best chance to win. It didn't matter who we picked, we were going to be a playing a quarterback who hadn't taken a live football snap in two years."
Frost said Gebbia "initially took the news really well." By Monday, apparently, something had changed.
"Every single guy on our team, we're going to care about and want what's best for them," Frost said. "We want guys who want to be here.
"Tristan's a great kid ... We preach that we care about the kids. And if they make a decision that doesn't involve us, we'd be lying if we didn't help them. We're always going to do that."
Martinez is set to become the first true freshman in Nebraska history to start a season opener. He missed all of his senior season in high school with a shoulder injury but ranked 103rd in the ESPN 300 for the Class of 2018. He was long committed to Tennessee before the Volunteers fired Butch Jones and Frost took the Nebraska job.
"In my own mind, I felt that I was the starter," Martinez said Monday. "I really just spilled my guts for it. Obviously, Tristan did the same. But if you don't believe in yourself, I don't think anyone will."