I'm concerned that mere addition of a stereo power amp will not change this characteristic about my setup, but I'm not sure if the "brightness" is a product of the receiver's amp or something else (pretty sure it's not the speakers).
You are very wise to be concerned about this!
What speakers do you currently have?
Generally, if you want to change the sound, change the speakers!
We can get into lots of arguments here about whether or not the amp will change the sound, but very few (if anyone) will deny that speakers have a profound effect on the sound while electronics are comparatively subtle.
The exception to this rule are:
1) Some type of tone control or processing is being done to cause the problem - any mainstream company like Yamaha knows how to cleanly amplify the signal; to do otherwise is to distort the signal. So the amp section should be good. You should have control of any processing/tone control as well as a means to bypass it. So if your sound is too bright when you are playing in pure direct mode, look to your speakers for the fix!
2) Your speakers provide a load that is beyond the amp's ability to properly deliver the needed power. Unfortunately, some of the lower model AVR's (and, notably Yamaha) seem to be current limited. What this means is that when playing (especially loudly) multiple speakers or speakers with low impedance and problematic phase angles, the amp's power section cannot deliver the needed power to cleanly drive the speakers. If not, you will get clipping distortion that is generally heard in the higher frequencies and makes for a harsh sound. I don't know if your Yamaha is a lower level or not.
Borrow or buy a more powerful amp from somewhere that has a good return policy to check for situation #2 above. If not:
1) Consider your room. Stand in the middle and clap your hands. How much echo do you get. Worst case is hard, reflective floors (without carpet), lots of glass and bare drywall on the walls and bare drywall on the ceiling. All of these will reflect the higher frequencies more than the low frequencies and you will end up with an overly bright sound. Carpet, draperies, furniture, paintings (not with glass in front), and anything else that will absorb or scatter high frequency energy are your friend.
2) Trial alternate speakers
in your room - again take advantage of distributors with good return policies. Right now, many places are allowing until the end of January for returns.