He did it with his first QB, but hasn't done it since and almost certainly won't ever again. Locker rushed just under 10 times a game in 2009 and 2010. Sark emphasized last year that he's no longer comfortable with the QB running 10 times a game, because of the accumulation of hits and increased injury risk.
We saw this play out last year. That skillset was forced on him, but he stubbornly refused to use that strength of both Card and Thompson.
Sark is also on the record many times that he doesn't want his QB running, aside from a few situational runs. He's an anti-Herman here. He doesn't want his QB taking the hits, so running ability isn't going to be something he values in a QB.
https://www.on3.com/college/texas-longhorns/news/steve-sarkisian-philosophy-running-quarterback-lack-texas-longhorns-casey-thompson-hudson-card/