How filterless Class-D amplifier works?
Class-D means using some sort of PWM. What you're showing is just the way the waveforms of a switching bridge looks like, and, in particular, a three-level-PWM (see the 2nd picture in the answer for an example): Vout+ in your picture would be V(x) in the linked example, and Vout- would be V(y). Their difference gives the blue trace in your picture, and V(x,y) as seen in the simulation. What you refer to as &quot;time delay&quot; is the effect of applying such modulation (the two branches are switching differently). In the linked example there is also the source for an LTspice schematic, feel free to use it if you can and are willing.
Since you're also linking an audio class-D amplifier, then you should know that the vast majority of loads for such amplifiers are electrodynamic loudspeakers which, at their bare  minimum, have an inductance and a resistance in series (it's more complicated). Driving such a load with a switching signal will bring up noise, cause unnecessary heating (besides increased EMI), but mostly it will cause more distortions which, since it's audio, are not wanted. Even if the switching frequency is greater than the maximum audio frequency (usually 10 times, or more), there are harmonics that are not wanted and, in the case of high-range tweeters, they can be damaging.
This is why, usually, there is a filter at the output of such amplifiers, to attenuate higher frequencies and leave the audio signal, as much as possible. However, these filters are, again, usually, made of inductors and capacitors, and their size and cost is not cheap, so for some cases -- notably woofers and subwoofers -- the filter can be eliminated. The reasoning is that:

woofers and subwoofers have large values for the inductance (from a few mH, upwards), so combined with their resistance they can attenuate the switching frequency to a certain degree (though not as much as an LC filter)
the audio frequency that these kind of loudspeakers can reproduce is in the low band, so their mechanical properties makes them quite immune to high-frequency noise.