Physics Question: Why does cabinet volume matter so much to bass response?

Well, it is a marriage between the driver and its abilities, and the cabinet made for it. You play to that particular drivers' strengths. In general, the bigger the woofer the bigger the cabinet. Again, this is applying to sealed and bass reflex/ported designs.

 It isn't a matter of the woofer being able to play louder any more than the woofer is supported by the cabinet's extension of resonance that aids in the woofer's lower octave response.

  You want big low bass, you will be the most successful with a larger cabinet and driver, given a proper design. And then you will move air.

You have to brace the heck out a large cabinet though, or else it will vibrate too much and the bass will be crap. A lot of smaller subs with strong cabinets can yield some amazing results.

Newer designs like the Devialet Phantoms manipulate Hoffman’s Iron Law using DSP and crazy amounts of power to get bass in small enclosures.

By the way, that was a very good question OP!👍

The simplest answer is that the air in the cabinet forms part of the suspension.

Before box speakers everything was pretty much close to open air/open baffle designs.