No class B amp is devoid of crossover distortion, but it can be made so low as to reach the noise floor at 1kHz, which is effectively zero distortion. At 20kHz its much harder to do this and usually the crossover distortion is distinctly measurable, though less than 0.001% is achievable. CFP output stages are by far the worst for this due to switching distortion - there is no way to add a speed-up cap as with an EF stage, so crossover artifacts at full power 20kHz are usually pretty obvious. However your tweeters have fried at this point, so you'll not be too worried about this!
Practical signals fortunately don't probe the HF full signal swing capabilities of an amp. You can measure this easily with two-tone IM measurements, say 18k+19k (dummy load only, disconnect those tweeters before doing this!). If that test shows low distortion you can be pretty confident that crossover and switch-off and large signal distortions are under control.
The best performers at this extreme can be MOSFET amps as switch-off speed isn't an issue at a lowly 20kHz. Bipolar EF stages too can perform well here with fast output devices, but CFP stages will show shortcomings due to switch-off.
As regards class B (50% conduction each device) and AB (> 50% conduction each device), its confusing because some people define an optimally biased class B as AB and others don't.
Whatever you define it, high bias moves the distortion to higher signal levels, increases THD in the process, but at higher power levels predominantly.
Music signals contain lots of spikes so the notion of the first watt is pretty bogus - quality signals of about 1W (not highly compressed stuff) will have spikes well up into the 10's of watts or more, so moving the distortion isn't going to help much, or may make it worse. Most (good!) music has a lot of dynamic range.
BTW the quieter the signal the less distortion matters because the distortion products will start to fall below the noise floor.
