The classic large-dead-zone audio amplifier, using two bipolars of NPN and PNP polarity with the bases tied together and to the servo-amplifier (negative feedback amplifier) output, exhibits a 2 * Veb = 2 * 0.7 volt or 2 * 0.8 volt dead-zone. The default UA741 opamp has 0.5 volt/microsecond slewrate; to slew those bases over the 1.6 volt dead-zone requires over 3 microSeconds.
This slewing time, when the opamp is only SLEWING and not able to respond to any input-output errors, is the duration of the impulse function your diagram shows injected at the output.
As bimpelrekkie suggests, change the input signal. I suggest you use a 20,000 Hertz sinusoid of amplitude 20 volts peak-peak. With 50 uSec period, and 3uSec dead-zone and 2 dead-zones per cycle, you should be able to visually notice the distortion.
Your question regards how the transistors "dampen" the distorton.
In truth, the loop is doing its job, the negative feedback loop is doing its job.
Increase the frequency to 50,000 Hz or 100,000Hz.
And inject some interesting "music" such as massed fiddles, with all those strings being tuned by the players' ears. Inject 1,000 Hz and 1,010 Hz and 1,020 Hz at the same time, and examine the FFT. Gather a 500 millisecond of samples, which provides 2 Hertz frequency-binning, so you'll clearly see the intermodulation.
The irritation of dead-zone during cross-over events comes from the wide-band nature of the IMPULSE function (very narrow in time) model of cross-over.
The ear is hearing spikes in the music, spikes that are not in the original program material, spikes that are dependent on when various sinusoids interact to generate a zero-crossing.
Why does THD not capture this phenomena? Because using a single-tone test results in the cross-over error spikes occurring only at the zero-crossings, and all the energy is harmonically related to the single-tone.
Multi-tone tests bring out the non-harmonically-related hash (randomly timed) spikes.