Many flathead utility engines still manage to crack the block around the exhaust port or pull out the head bolt next to it. Iron Kohlers come to mind and Briggs to a lesser extent. Also, flatheads have a big chamber that sends a lot of combustion heat into the cooling "system," making them more thermally inefficient than an OHV. They have an inherently large chamber volume as well as the valves being on the far side it, both of which conspire to decrease volumetric efficiency, burn quality, and, dare I say, increase exhaust emissions. They do have advantages, but I don't think they exceed the advantages of an OHV.
I think it's interesting that pushrod designs may actually predate flatheads in the evolution of the ICE and that they obviously superseded the flathead as well.