Given enough time, BCH will have the same issue with block sizes filling up.

The hard coded block size limit is 32 MB in Bitcoin Cash. This may prove to be an issue at some point in the future - and exponential growth of the block chain may indeed become an issue. At the other hand, given enough time, there are lots of optimizations on the BCH roadmap reducing the implact of bigger blocks - and if Lightning really does prove to work out, there is no problem to roll it out on Bitcoin Cash, too.