should a music company have the right to have a news podcast removed on copyright grounds, when it's not even clear that said company has had any copyrights violated?
Should they? No. But the DMCA gives them the right (or at least the ability) to do so. It gives it to you, too. My understanding is that anyone can file a DMCA takedown notice.
I have often wondered what would happen if people started filing DMCA takedown notices by the millions on major websites against the big content producers. There doesn't seem to be any penalty for filing bogus notices.