2. Brian Griese, 3rd round, 1998 NFL Draft
Okay, this one is slightly cheating, I will admit. Brian Griese was drafted in John Elway's final year with the Denver Broncos, but we're going to move the goalposts a little bit for him because the Denver Broncos simply haven't drafted many quarterbacks since John Elway retired, not quarterbacks that have actually played for the team, anyway.
Brian Griese was picked in the 3rd round of the 1998 NFL Draft out of Michigan, and things didn't get off to the best start for him with the Broncos as he threw an interception in one of his first three pass attempts with the team as a rookie.
He got the opportunity to take over as the starter for Mike Shanahan's squad in 1999, the first year of the post-John Elway era, and it was a rather disastrous year for the Broncos, all told. Not only was Elway's career over, but the 1999 season was when Terrell Davis suffered a major knee injury that limited him to just four games that year and ultimately proved to be a career-ending injury. Griese was supposed to be able to lean on TD and that Broncos running game, but that didn't ever happen.
The Broncos were much better in 2000 with Griese at the helm, although he was limited to just 10 games that year. Despite playing only 10 games, Griese made the Pro Bowl after throwing 19 TDs compared to just 4 INTs. The Broncos would go on to have a top-10 offense in three of the four seasons Griese was their primary starter, but that wasn't enough to keep the team a playoff contender, nor was it enough for the Broncos to pass on the chance to add Jake Plummer in the 2003 offseason.
Brian Griese lasted five seasons with the Broncos, starting 51 games in total. Griese, despite helping lead the Broncos to a trio of top-10 offenses, never played a single playoff game for Denver or any of the other teams he played for. Even with that, he's still got a Super Bowl ring as a member of that outstanding 1998 Broncos roster.
Griese is now a coach on Kyle Shanahan's staff in San Francisco. It's a small world.