Yes, the triode operation of an amplifier that has tetrode or pentodes is compromised. Such tubes in triode mode are not as linear as real triodes.
In triode mode, a push-pull amp still differs from a triode amplifier that is single-ended. Push-pull has greater power and wider bandwidth, but also *more* distortion at low power levels. SETs have vanishingly low distortion at low power levels, on account of the way the output transformer behaves when operating single-ended (no hysteresis loss) as opposed to push-pull (where hysteresis loss causes a loss of low-level detail due to increased distortion).
Of course there is more but that's a good place to start...
In triode mode, a push-pull amp still differs from a triode amplifier that is single-ended. Push-pull has greater power and wider bandwidth, but also *more* distortion at low power levels. SETs have vanishingly low distortion at low power levels, on account of the way the output transformer behaves when operating single-ended (no hysteresis loss) as opposed to push-pull (where hysteresis loss causes a loss of low-level detail due to increased distortion).
Of course there is more but that's a good place to start...