In the vast majority of cases you never want to push a solid state power amp into distortion, they hardly ever sound good and often "latchup" and make clicky or farty noises.
That is not to say a decent overdrive tone cannot be wrought from solid state but it all comes from the pre amp and makers use various distortion mechanisms such as clipper and non-linear feedback using diodes, FETs and transistors.
The "pushed speaker" idea is a bit of a myth as well. The horses mouth, Celestion, have said that you do not have to drive their speakers to within an inch of their life to achieve a good tone, if the drive signal is right, milliwatts sounds great!
Indeed, in a valve amplifier especially, it is important that the speaker(s) is rated to at least 50% over amp rating for a long life.
Pedals: Well there is at least one range (cough!) that uses a double triode in a proper 300volt circuit.
Dave.