OK, so how many coaches have made it to the AFC/NFC championship game, in their first four seasons, just during the Texans history... hmmm
Jon Gruden, Andy Reid, John Fox, Jim Mora, Lovie Smith, Tom Coflin, Mike McCarthy, Ken Whisenhunt, Sean Payton, Brad Childers, Jim Harbaugh, Pete Carol, Bruce Arians, Dan Quinn, Bill Callahan, Tony Dungy, Norv Turner, John Harbaugh, Jim Caldwell, Rex Ryan, Mike Tomlin, John Fox etc... etc...
Is that relevant enough?
Well of course, it's like you always find it the last place you look. You quit looking once you find it! All of the teams you mentioned made it to an AFC/NFC championship within the first four years of their respective coaches. How many of those teams would have let a coach stay if they had a losing record after 4, 5, 6, 7 years? Especially if they played in the weakest division in the league for most of those years. If you go across the league, you can see it loud and clear, the vast majority of teams keep looking until they find a coach that can take them deep into the playoffs. Some never do, but they don't give up for the sake of "stability".
This ain't college bro where a program can dominate for decades due to money, recruiting, facilities, loyalties, reputations etc... this is the NFL which is based on parity. A salary cap, a draft, scheduling etc... all designed to give advantages to weaker teams. This is the league built for quick turnarounds. Your college analogy makes no sense at all. How often does a bad college program turn around and be ranked in the top four in four years.... let's see, like almost never. Could I name 50 college teams that haven't been ranked in the final four in the last 30 years. Well 31 of 32 NFL teams have made it in the last 30 years! Your college analogy proves changing coaches in the NFL till you find a good one is the way to go, because a coach in the NFL can make the difference quickly so much more often than in college. I gave you over 20 examples why.