I agree its a bit over the top to take it all down. But how many non copyrighted things do you find on those filesharing sites vs public domain files. The problem I personally have with those sites is there is no accountability to the person that shared the illegal files. NONE - NADA - ZIP - ZILCH. The most I could do to stop a file from being shared was to report it to the host, they would ALWAYS promptly delete them but then they would just get shared again within a few days (usually)
That dude (or dudette) that shares the file has nothing to lose and total anonymity.
A much better way to fight the problem IMHO is to take that away. For example make them have a REAL account verified with a $1 charge for a lifetime membership via credit card or something similar. If the person is sharing legitimate files then hey no problem, nothing to worry about. I would guess that people that share out hundreds of paper models or songs or movies or whatever else they desire, had their account verified to a real person (themselves) they would be a little more hesitant in what they would share. Well that or their parents would be pretty irate when they got a call....
once again purely my opinion...