The emphasis on QB mobility will also end very quickly when a Mahomes or Fields or Lance suffers a horrendous injury on a designed QB run. I know people like to say things like Lamar Jackson is able to avoid big hits, but no one with a ball in the NFL is able to prevent getting blown to smithereens every once and a while.
Mobility and designed running are very different things.
Designed runs are a very specific thing.
Ability to escape and run for a 1st is usually what more people mean by mobility.
The next level down in mobility is simply... the ability to evade the pass rush, and avoid hits.
Immobile QBs take a lot of hits that mobile QBs don't.
Here are the bottom 4 full time starting QBs in hits taken last year.... Lamar Jackson, Kyler Murray, Baker Mayfield, Aaron Rodgers, Cam Newton
Here are the top 4 full time starting QBs in hits taken last year: Cousins, Ryan, Herbert, Wilson, Tannehill
So sure, designed rushing QBs are going to take hits, but that is a different issue than mobile QBs vs. immobile. I wouldn't consider Rodgers a rushing QB, but he is mobile and he greatly decreases his hits taken that way.