It's plausible, it's logical.
However, when a team finds a QB they can win with. That equals job security for everyone in the building. The GMs and HCs want to win the whole thing, sure. But they want to keep their jobs more.
So they choose the typical path. Pay the QB and just hope you can find a way to draft well to make up for it. Even if you don't draft well and it goes south, a Franchise QB can still get you to the playoffs. Which in turn keeps you employed for years.
Finding QBs is harder than one would think. Only about 20% taken in the 1st round go on to be Franchise QBs. 80% failure rate. Only a few over the last 10 drafts which has had over 120 QBs taken in that time frame have been fringe/franchise caliber taken in the 2nd rd or later.
You can't just go in the backyard and pick a Franchise QB off of the Franchise QB Tree. If it was that easy every team would just do that, and no one would pay a QB. "Just draft another one."
Imagine the Eagles trading Hurts because they don't want to pay him. And struggle to find his replacement and become losers. Everyone is getting fired. Thus he is getting paid, and they take the typical path. Jobs are secure.
The problem with that typical NFL team plan, is that, just getting to that place of worrying about job security requires finding that exceedingly rare franchise guy in the first place. Everything from winning to keeping jobs is contingent on landing that super star.
The number of people who might be at risk of losing their jobs, by dealing a potential franchise QB, pales in comparison to the number of people who are constantly losing jobs due to wasting draft picks and cap space chasing after, and failing to land, that one guy.
The whole point of my premise is that you never have to find that rare, true franchise QB. You just have to find an average-good QB who can be plugged into a true all-star team.
If you happen to get lucky, and hit on a late round super star, then sign him to an extension with low cap hits in the first three years, and trade him for a boatload of picks. How many firsts do you think the Eagles could get for Hurts right now?
Use those picks to stack your roster, and use the freed up cap space to fill critical holes with high-dollar vets.