The GM LS-family of engines is exceptional, and they use pushrods. It was an attempt to see how much modern design could keep pushrods relevant, and the results were good enough to continue pushrods for quite a while.

The main benefit is they cost less than OHC, especially if you are only using 2-valves per cylinder. They also allow a shorter engine, but that's only a benefit for something like a Corvette, while millions of SUV's from GM didn't care if a pushrod engine was an inch shorter.

I would go so far as to say that if you are adding a turbo or a supercharger, you only need 2-valves per cylinder. For naturally aspirated, OHC and four valves per cylinder seem to be dominant, so there must be a benefit to that.