Where is this unicorn you speak of? Three of our last four "running qbs" had one to two great games then were less than average in comparison to our "throwing qbs". ZB was at or above average production for the "throwing qbs" but he's also coaching in the NFL but he still had injury issues. Injuries are an accelerated issue. Manzel, Tebow and gooner smurf are about the only three qbs with any kind of longevity without injury issues. Vince Young and Colt McCoy even had injury issues in college. Your odds of finding the unicorn are extremely low. It's a wonderful idea that just doesn't pan out.
Nearly every current successful NFL team is leverage mobile QBs. Below is a list of teams that made the playoffs in 2022. Interestingly, all of these same QBs played in college as well, so they exist. I would say out of the (14) teams listed below, only 3 or 4 of them utilized primarily pocket passing QBs.
AFC
Chiefs - Mahomes - super mobile, elite pocket presence
Bills - Josh Allen - running QB / dual threat
Bengals - Joe Burrow - above average mobility, good pocket presence
Jaguars - Trevor Lawrence - true dual threat
Chargers - Justin Herbert - above average mobility, good pocket presence
Ravens - Lamar Jackson - running QB / dual threat
Dolphins - Tua Tagovailoa - above average mobility, good pocket presence
NFC
Eagles - Jalen Hurts - great mobility / dual threat
49ers - Lance (running QB/injured), Garoppolo (statue/also injured), Purdy (highly mobile, great pocket presence/also injured)
Vikings - Kirk Cousins - limited mobility, but good pocket presence
Buccaneers - Tom Brady - least mobile QB of all time, but results speak for themselves
Cowboys - Dak Prescott - great mobility / dual threat
Giants Seahawks - Geno Smith - adequate mobility / average pocket presence
Point being - the game has evolved to smartly utilize mobility from the QB position. Just because a guy can run doesn't mean they necessarily increase risk of injury. I'll take a guy like Trevor Lawrence, Joe Burrow, Tua, or Mahomes who are pass-first, but know how to get chunks of yardage with their legs when it makes sense.