Think of it this way: the plate of the tube is not feeding the output transformer, but rather the output transformer is feeding the B+ to the tube. The output transformer (or half of it) replaces the plate resistor that you see in a preamp stage. No cap there, so no cap on the OT primary.
The signal getting to the secondary of the OT is happy side effect of the power tube's changing conductivity (controlled by its grid), and changing the amount of current flowing through the OT. There's no current flowing from OT primary to secondary, if that's what you were thinking.
As far as blocking DC through the tube itself, it would be hard (understatement) to bias the tube with no current. I'm not sure if that would work in class C or not?