My ShopFox 1704 has a 3/4x16 spindle, MT-1
It's not unusual for barrel trimmers to peel away some of the brass tube. If that's causing your tubes to break free, the problem is in how you glued them into the blank, not with the trimmer.
A very common problem is for the trimmer to trim away a very small portion of the outer OD of the tube, but leave a microscopically thin inner tube. What happens is that the trimmer removes a portion of the brass, but because the trimmer blade doesn't fit the trimmer shaft tightly, a portion of the tube is left. Not a problem - just use an ordinary knife to trim away the excess tube.