And why do they demand fees paid from venues where performers play only original music?

They would claim that it is extremely unlikely that a venue where music is performed will be absolute in NOT performing published material. If a venue made that argument, all ASCAP or BMI would have to do is sit in for every show until one performer sang one published song (Happy Birthday) and then they would have all they needed for a law suit.

Most venues back down and pay the rate.