You can't begin by saying that they have had no influence at all and then immediately go on to say that they have "vulgarized the debate" so much that politicians wouldn't even dare touch the issue. That's kind of impossible.
I think you are wrong on both points. Both the green party and the left party now have stances on copyright that are very similar to the ones that the pirate party have propagated. The pirate party were also quite visible in the media at times, although I'm sure they were ignored by many.
I think that the public debate before the entry of the pirate party (and the pirate bureau) was one of "I want free stuff" against "we need to pay the artists", after their entry it turned more sophisticated and idealogical. I think this is a good thing.
Of course, there are also complete nuts in the party, and some of the arguments put forward aren't particularly good. I never really liked the former party leader.
However, Christian Engström
seems to be doing some good down in the European Parliament right now, and as a force against what I perceive to have been a very one-sided "we need longer copyrights, and we need to jail everyone as soon as they copy one song!!!1" discourse I think they have been a very valuable force.