I can't comment on the issue of comparable numbers between dirt and grass racing, but there is a larger historical issue that might explain why European horses are actually better on the grass.

The biggest stables in Europe, led by the Arabs who race there, started buying the very best stallions active in the US during the 80's (and a little before and after that decade.) There were some sales at which they seemed to buy EVERY Northern Dancer horse and broodmare available.

They sucked the juice out of US breeding in general and turf breeding by default. The Japanese also stepped in in a big way and might have pulled down the next tier of great stallions.