The amplifier does not need to misbehave at higher frequencies to cause problems. Non-linearity simply increases at higher frequencies producing mixing products which may end up in the audio band.
That's pretty much what I meant by misbehaving and is probably one of the reasons why many audio amplifiers are designed to work well above the audio band, so ultrasonic noise from CDs and other digital sources don't cause buzzing. For example: the LM3886 is specified for low distortion up to 20kHz and has a full power bandwidth of 80kHz and the NTE1380 has a FPBW of 140kHz. Somehow I doubt they're going to produce audible noise from 44.kHz.
FPBW (full power bandwidth) is just another name for large signal response and depends on slew rate and output voltage. The gain-bandwidth product or small signal bandwidth is more important for reducing distortion unless the full power bandwidth/large signal response/slew rate is insufficient which should never happen.
The NTE1380 datasheet does not show a schematic but the LM3886 does. See those 1.1k resistors used for emitter degeneration of the input PNP differential pair? They lower the input stage transconductance while maintaining the same tail current. This allows the compensation capacitor marked as 10 picofarads to be smaller then it otherwise would need to be improving the output slew rate and full power or large signal bandwidth. As a side effect, those 1.1k resistors in series with the emitters increase the input noise considerably. If a schematic is not available or is incomplete, then the input noise specification can be used to determine if emitter/source degeneration was used. Contrast the 318 and the 833; the former has 10 times the slew rate (and full power bandwidth) at the cost of 3 times higher voltage noise do to emitter degeneration to reduce transconductance.
The old 318 operational amplifier used emitter degeneration resistors to get its high frequency performance and was noisy and lower precision as a result. Most audio power amplifiers do as well. FET input amplifiers have lower transconductance to start with. There are also other ways to handle transconductance reduction.
For audio power amplifiers, input noise is not normally a concern because it is assumed that a high level signal is available but I have noticed that many modern amplifiers produce a really annoying hiss with no input signal when connected to high efficiency speakers.
The excess gain at frequencies above the audio band is just a side effect of maximizing the gain at high frequencies to limit distortion. I do not think anybody does it to limit intermodulation of content above the audio band which should not exist anyway and would not have been a problem before digitally sourced music. But I do wonder if early inconsistent reports of digital audio quality came about because of inadequate anti-alias filtering allowing high frequency alias products to mix in the power amplifier. The anti-aliasing filters were an obvious place to save money and lots of cheap audio gear did so.
It's true a good filter will be required to block the higher 44.1kHz harmonics, which could do something funky, but I doubt the fundamental needs to be attenuated to ridiculously low levels, to avoid noise.
Burr-Brown and others were not recommending 11th order anti-alias filtering with a transition band between 20 and 22 kHz to make their 44.1 kHz ADCs and DACs easier to sell. This was such a big problem that oversampling ADCs and DACs with their much simpler anti-aliasing requirements replaced them quickly and made this a solved problem even on the economic side.