Motorcycle.com has an article on the 2010 Triumph Thunderbird, it speaks of the hard downshifting, I have not read anything on the heavy duty clutch/transmission anywhere. I imagine that it has to be strong considering the torque and horsepower that these engines are designed to put out.
I understand that some Triumph parts may take a week if it comes from overseas, domestic parts supplies may take a few days. Your idea of calling is appropriate considering that it´s your bike, you wish to stay informed without being pushy and you care. Your bike is not the only one there, the part may arrive yet they are finishing another bike started earlier this week. I´m curious about the reason for the shimmy since it started after the tire change and the belt is so premature.