Likely culprit for harshness at higher volumes?

Add a subwoofer for better foundation.
You are probably boosting the volume to get more bass
and this is overdriving the other frequencies.
This used to happen to me.

I am turning up the volume a bit; not necessarily to get more bass but to hear some of the other instruments that may be in the background and/or a little quieter.

I found that on some CDs, at background listening levels, sound seemed to come mostly from one side or the other. When turning up the volume a bit, I'd then notice additional instruments coming from the other side (mostly some sort of percussive instrument). Since then, I've gotten used to playing CDs at a certain volume levels.

So, the 'desired' listening levels may be a little loud but I don't think they are overly so -- I'm not concerned about damaging my hearing at these levels (but perhaps my ears are already bad :)).

But if certain frequencies are being overdriven, what's the solution? Get different speakers? If so, any specific speaker attributes I should be considering? Larger tweeter / mids? Bi-wire capable? Capable of higher power inputs?

Or is this an amp issue? Need a more powerful amp?

Sounds like (excuse the pun), it's an issue with the amplifier. Your Anthem is a pretty good pre, but Adcom power amps are not that great (having owned one). Seems like at low volume the adcom lacks detail and dynamics - you need to crank it up to get anything out of it. It may be having trouble driving the 4 ohm Kefs. You may not necessarily need more power, just a better quality power amp.
Best idea would be to borrow or try another/better power amp and see what happens. A good power amp will have lots of detail/dynamics at lower volume levels.

hat on some CDs, at background listening levels, sound seemed to come mostly from one side or the other. When turning up the volume a bit, I'd then notice additional instruments coming from the other side (mostly some sort of percussive instrument). Since then, I've gotten used to playing CDs at a certain volume levels.

You may have a burnt voice coil that is rubbing - possibly one or both speakers are damaged. What you describe is definitely NOT normal.