Interesting read on stats of QBs since 2000 that have started right way, weeks 2-8, weeks 9-16, year 2, year 3, or year 4+
https://www.cbssports.com/nfl/news/history-of-nfl-qbs-waiting-to-start-heres-what-stats-since-2000-say-about-when-theyll-most-likely-succeed/
Between 2000 and 2019, there have been 91 quarterbacks that entered the league and became a team's full-time starter at some point. (That's an average of around 4.5 per year.) It may have happened right away, or it may have happened a few weeks or even years down the line. It may have happened with the team that drafted them or it may have happened with another team entirely. But at one point or another, they all got a shot to prove themselves.
Of the aforementioned 91 quarterbacks, just 29 of them started under center in Week 1 of their rookie season. An additional 21 took over the reins at some point within the first eight games of that season, while 14 more grabbed them in the back half of their rookie year. That's 64 of the 91 passers who became the full-time starter during their initial season. Of the remaining 27 players, 16 took the job at some point in Year 2, three took it in Year 3, and eight had to wait until Year 4 of their career or later.
In almost every category, the group of players that took over in Year 3 performed the best, while the players who became starters in the second half of their rookie season performed the worst. Of course, these results are subject to all sorts of small-sample size concerns.
You can't just look at the Year 3 column and see that those players have significantly out-performed the others and conclude that the correct move is to sit your preferred starter for two seasons and then give him the job. That group consists of only three players, two of whom (Rivers and Romo) went on to become high-level starters for a decade or more. Similarly, you can't just look at the Game 9-16 column and conclude that you should never give your guy the job during that section of the season because those players haven't been as good. It's entirely possible that players like JaMarcus Russell, Charlie Frye, Drew Lock, and more were going to fail regardless of when they stepped into the lineup.
It does seem notable, though, that the rookies afforded the opportunity to start right away have generally out-performed those who had to wait a bit but still stepped into the lineup in the first half of their rookie season, who in turn out-performed those who had to wait until the second half of the year before their team gave them a shot. It's also worth noting that while in general, the quarterbacks who waited until Year 2 or beyond to become the starter have out-performed their rookie-starter counterparts, the former sample has the benefit of including Brady, Mahomes, Rodgers, Brees, Rivers, and Romo.