Reverse-engineering my speakers' API to get reasonable volume control

This begs the question, why buy oversized speakers of which you can only use 10% of the range instead of smaller speakers?

Is the sound quality better when not approaching to maximum volume?

You can't buy quality sounding speakers without them being high powered.

Do you mean physically, or as in there's no market for high-quality low-powered ones?

Both. To fill a larger room with enough balanced sound with all types of music implies enough headroom that most of the time it will be overpowered. Also everybody would take a mostly overpowered speaker over a sometimes underpowered one.

The "small quality speakers" category is filled by decent bluetooth speakers and a few pc/desktop 2.0 models.

These are complexly related parameters. All else being equal:

- a larger enclosed volume produces lower frequencies more efficiently

- a larger Xmax (range of motion) on a conventional driver produces louder sound

- a smaller Xmax produces less distortion

- a larger driver area produces louder sound

- a larger driver area can produce more distortion unless the engineering compensates for that, which usually involves more expense in materials and assembly

- the more efficiently coupled a driver is with the room air, the less distortion results

- there are several ways to improve the coupling of a driver to the room air, all of which increase one or more of cost or complexity or size

- larger and/or more complex cabinets cost more than smaller and simpler ones

- adding drivers to a speaker increases complexity

If we had perfect materials, we might build a single driver speaker made of infinitely stiff material of zero mass that accelerates/decelerates infinitely quickly in response to exactly enough power to overcome air pressure, and the parameters we would still have to tweak would be size (to control maximum sound pressure level) and whether we wanted to place it in front of an infinitely sound-absorbing back wall (to change it from an omnidirectional source to a cardioid source).

We don't have any of those things, so for a given budget of price, size, parts availability and complexity we need to make tradeoffs to get to our targets for loudness and quality; not all targets can be reached from a given constraint.

TL;DR: yes.