What’s the relationship between gain (dB) and power (watts)?

With all the math and physics measurements being tossed around, you’d think that everyone was equally gifted with identical hearing and processing by their brains. Unfortunately, this is not true and varies again by the condition and training of the listener and by their age. This is what makes "audiophiles" out of "just another person, but gifted by the Lord with perfection and the ability to communicate perfectly." I hope you are laughing at how arbitrary the listening experience can be. The most important aspect of the whole thing is how much you enjoy the experience of recorded sound and music.
I have a degree in experimental psychology and have been an electronic design engineer and owner of a graphics/media company with a recording studio for over a decade. This is irrelevant as I ENJOY music and have for a long, long time.

I am interested in this because I have just upgraded from one power amp to two monoblocks - the gain has not changed but the available power has gone from 200W per channel to 600W...

... and yes, the monoblocks are no louder than the stereo unit. 

Here is my attempt at understanding the issue. Typed out here more to get feedback, to get my working checked, than to tell anyone else what to think.

OK, so gain tells you how much bigger the output voltage is to the input voltage, i.e. the ratio of input to output. It think (guess) that typically power amps have an input to output voltage ratio around 100 times. 

Power considerations appear to come into play when you connect the amplifier to something, like a resistor, or a speaker. In order to hold the correct voltage, the amp has to pump out current. If it runs out of current, the voltage will decrease, the relationship between output voltage and input voltage will falter. 

With something simple like a resistor, one could easily work out how much power one needed from power = voltage squared divided by resistance. But speakers are not resistors, and music is not a steady state. I think that if one suddenly wants a speaker's driver to move, its resistance momentarily collapses, and so only an amplifier with a lot of power can hold its output voltage steady. As the speaker resistance collapses due to a rim shot or square wave, the amplifier has to dump a tonne of current into it.

In my mind big power amps sound powerful because they are capable of making the speaker follow the music. They sound more exciting, not louder. 

Wouldn't an increase in the gain setting be the equivalent of turning up the volume knob, as a default setting? (using more power by default?)

I have an amp with independently adjustable gain for each of 4 inputs. I don't use it, but my understanding is that it will simply adjust (offset) the default volume (power) level for a given input. This makes sense if you wanted to level out some inputs that might be varying in volume, but why the global gain switch? Gives you a better/preferred starting point for volume.. due to input or speaker behavior?

@rols

In my mind big power amps sound powerful because they are capable of making the speaker follow the music. They sound more exciting, not louder.

I thought that increased damping factor was part of the reason that higher-power amps have more speaker control (grip), but I'm probably missing other factors; interesting topic.

This is where my technical ignorance seems significant.  Maybe I’d need a textbook to understand this:

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails.

This is nothing more than an unnecessarily techno-jargon laced way of saying exactly what I said: Gain is a multiplier, power is a ceiling.