What actually determines volume power? Is it watts?

For what it  might be worth and given that our systems are VERY different, here is my experience.  I have a 75W tube amp (ARC Ref75se using 8 ohm tap) driving speakers with a nominal  impedance of 8 ohms and a sensitivity of 91db-listening room (and 89db anechoic chamber).  I have never seen the meters on the amp ever go higher than 5-10W.  I usually listen with the volume setting on my preamp (0-103) (ARC Ref6) between 3 and 26, background and dedicated listening when I want to blown away so that nothing else exists.  I have very "transparent" (NOT the brand) cables.  Anything louder borders on painful and I am afraid of blowing something out.  There is no where in this system to adjust the gain.  I wish there was because I'd like to use less attenuation of the volume.

With respect to volume, lots more in the signal path matters. I do not stream. Source: caching server with its own internal SSD, Macbook pro running JRiver Media Center, hi-res files on computer or external SSD, and embarrassed to reveal you tube.   I run only NOS (no over sampling, all processing is disabled in JRiver except volume leveling and PEQ - parametric equalizer).  Volumes for each source are different.  All sources run without attenuation.

Conclusion, to echo what many others have suggested: this needs a systems approach.  The best solution to OP's original question would be found in the synergy of the individual components of the system as a whole.  And for ultimate endgame satisfaction the solution must include the often forgotten details that matter like the room's dynamics and the listener's hearing, sonic & music genre preferences.  It has taken a long time to sort through all this for myself.

SPL which answered all my questions (and is fun to play with) Toptes TS-501B from Amazon for less than $30.

Happy listening dear friends, Leon AKA Timbrehead

I agree with the comments that there is an equation here: amplifier output / speaker efficiency, and that speaker efficiency usually matters more. 

My Altec 604Cs are 101db, 16 ohms. That’s way more efficient than most stuff being made today. 

The question on if the OP needs more watts to drive more volume is an easier question to answer than I first realized and touched on by some but not specifically called out.  The Integrated Amp has fully functional power meters.  At 70 % volume, what is the amp putting out per channel?  If it’s a ton of watts, 70 % of the 100 watts then looking at a different amp might make sense.  I’d guess it’s far lower and the amp can put more out if the volume is cranked further.  If that still isn’t delivering the volume the OP is looking for, then really honing in on the speakers sensitivity and amps power supply / amperage along with watts, synergy with all the gear in the chain, specific attention with matching the gain levels of the gear would be the way to go.  Steve Huff did an extensive review, tried at least 3 different speakers with it, concluded the 3200 would likely drive 95 % of speakers out there.  He concluded it’s not a beast like a Coda, Krell, Pass but should be able to drive most speakers. 

The "Stereophile" review of the Yamaha AS-3200 pointed out that this very fine integrated amplifier should not be used with loudspeakers that present a load of 4 ohms or less.  That makes me wonder about the loudspeakers that you are using. Otherwise, many other contributors have offered some good advice.

I hope you sort this out.  The AS-3200 is a wonderful piece of hi-fi gear.

you need a new system, from intergrated to pre+power amplifier