What actually determines volume power? Is it watts?

Here's an opposite story.  I have an old Krell stereo amplifier, pure class A, the KSA80.  It is rated at a mere 80-Watts per channel into an 8-Ohm load.  It is a big beast, I struggle to pick it up, and it has been rated as one of the 10 most influential amplifier designs of all time.

But it is also rated at 160-Watts per channel into 4-Ohms, and 320-Watts into 2-Ohms.  Each time the load halves its impedance, the amp delivers twice the current, and Watts is current times voltage drop.

I now use it with KEF Reference 1 speakers.  These have a sensitivity of 85-dB for a 2.83V signal at 1-metre distance, so quite low efficiency.  Nominally 4-Ohm speakers, they drop to 3.2-Ohms.  I have NEVER hit my personal volume limit and I play music loud.

Pushing the lowest bass out to a powered subwoofer obviously relieves the main amp.  I have a class D subwoofer rated at 1250-Watts RMS, or 3000-Watts peak power.  It has an input level selector and I run it on level 3 out of 60.

When looking at amplifier specifications, they tell almost none of the real story.  Look at the amount of distortion they claim at the rated power. Is the rated power with just one channel driven? How much does it drop when another channel is driven?  Droop here tells you the power supply cannot keep up.

Is the power given into 8-Ohms, or maybe 6 or 4-Ohms?  If it is for 4-Ohms, expect it to be only half that at 8-Ohms!  Does the power double each time the load impedance halves?

Almost all speakers have nasty dips in their impedance in crossover regions.  There is no such thing as a speaker that is 8-Ohms right through its frequency range.

If your speakers don't play loud enough, you could try sitting closer ... and I could keep banging on, but I won't.

@dman

If, as some suggest, you look at changing both your speakers and your integrated amplifier, could I suggest you look at powered speakers where the manufacturer has effectively done the matching for you?

The little KEF LS50s are the knockout class-leader in their price range and play much, much louder than their appearance suggests. Each active speaker includes a 260-Watt class D amplifier for the bass and a 100-Watt class A/B amplifier for the treble.  So you effectively get bi-amplification and monobloc operation.  A bonus is that these units have concentric bass and treble drivers, so they throw a coherent wavefront resulting in a massive soundstage and a much bigger than usual listening area.

There are much more expensive powered speakers in the KEF range worth considering as well.

As others have said, one important role of the power amplifier is to control the momentum of a dynamic speaker, so it actually stops moving in short timeframes.  Eliminating speaker wires allows the amplifier feedback circuit to directly interact with the speaker