Why is power gain a meaningful quantity at all?
The gain of an amp in decibels is never voltage or current, it is always power.

Nope. A 30 dB power gain is a 30 dB amplitude gain. See: how decibels are defined for amplitude and power quantities (namely: differently, so that this works).

But aren't we usually more interested in how much output voltage we are getting from the amp?

That fully depends on what you're doing, so I'll go with a &quot;no, not usually&quot;.

The amplitude of the speaker vibration is not really a function of how much power is dissipated in the speaker (let's assume a simple dynamic coil speaker),

of course it is!

but rather how much current is flowing through it (right?).

which is the same thing, but square-rooted (and proportional).
Also, for you as human listener, the amplitude of the membrane movement isn't more important than the power; both are linked inherently, and with human perception being logarithmic to intensity, whether or not we square the current or not makes only a difference of a constant factor. So, you're wrong to put emphasis on that!