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.

It's possible that your speakers have a distortion which is present but below the detection threashold at lower levels, and then as you go up to high levels, the distortion becomes louder than the decection threshold. In other words, your ears may have a non-linear perception of what is fundamentally a linear distortion.

It is also possible that your speakers have a distortion or coloration that is level-dependent, or non-linear. For example, if the voice coil of your midwoofer is heating up faster than the voice coil of your tweeter, then your midwoofer will compress more at higher levels. So you could end up with a brighter tonal balance at high levels.

My instinct is that the first situation is the case, this based on my work with horns. In effect a Uni-Q driver horn-loads the tweeter, but much about that "horn" is not optimum from a distortion-minimization standpoint. We see this in prosound with a sub-optimal horn: It sounds fine at low levels, but crank it up and it sounds harsh. The distortion was there all along, but we don't hear it at low levels.

Imo, ime, ymmv, etc.

Duke
dealer/manufacturer