Man, you've gotta have bad luck or something. The Tascam King (aka Cory) set up my tape path flawlessly and I did experience quite a bit of shedding but not enough to actually get between the tape and the heads.
On my Otari, which is a console-style 2-track that I use only with ATR tape, I have a little pile of brown shavings that I periodically have to clean off. ATR is kind of a dirty tape, but the quality of the recordings I've made on it make it well worth the extra maintenance.