Joseph said the choice is a "permanent" decision.
Brandon Stokley of 104.3 The Fan first reported the team had made the decision to go with Siemian.
"I am disappointed," Lynch told reporters after the announcement. "I have to be prepared. I believe in myself to be a starter. But coaches' made the decision."
The Broncos were heavily linked to former Dallas Cowboys quarterback Tony Romo during the offseason before he opted to retire in order to accept a broadcasting position with CBS Sports.
Yet, even before the 37-year-old veteran decided on a career change, Joseph made it clear he was confident in the options already on the roster, per Sheil Kapadia of ESPN.com.
"We've got two young quarterbacks that we feel really good about," he said in March. "One played a bunch of ball last year in Trevor, and Paxton is a first-round pick for a reason. He is a talented guy that's got a bright future. It may be now. It may be a year from now. But we have two young guys that we feel really good about."
So it didn't come as a surprise when the Broncos didn't make a major splash to upgrade at the NFL's most important position. The biggest change was picking Chad Kelly, the nephew of longtime Buffalo Bills starter Jim Kelly, in the seventh round of the 2017 draft.
In July, Colton Strickler of Mile High Sports noted general manager John Elway struck an optimistic tone about the quarterback competition heading into training camp and the preseason.
"I feel good about that spot," he said. "If you look at where we were last year, in my mind, we're in a much better position this year with Trevor having a year under his belt. Then, with getting Paxton in the draft last year, and him having played a couple games and having a year under his belt."
In the end, the Broncos decided Siemian gave them the best chance to win football games right now. While he finished the 2016 season with a mediocre 84.6 passer rating across 14 games, he did flash upside with an 18-10 touchdown-to-interception ratio.
Denver is built on a strong defensive foundation, so the quarterback isn't asked to carry the same burden as some other teams around the league, but consistency at the position is still key to success. That's what they'll be looking for from the 25-year-old Northwestern product.
Lynch, a first-round pick from the 2016 draft, may still get his opportunity to shine at some point during the year if Siemian fails to provide stability, though.