Likely culprit for harshness at higher volumes?

Tsk, tsk, tsk. Twenty one posts to this thread and no one has suggested upgraded power cables? :O
Seriously though, sounds to me like speakers or room reflections are the most likely cause of your problem. You are on the right track with your 4-step program. Although I would move number 4 up to 2 and scrap number 3. Good luck and keep us informed.

Saru, why don't you at least post the city or vicinity you are located. I am sure others may be willing to stop by and listen, diagnose, and even help you ID the problem with bringing some stuff (equip, cables, deflectors, etc. . . ) over.

Any component can create congestion, HF glare, etc. . . as part of a combination with other components (or alone).

I once owned an amp (Theta Dreadnaught II) that was literally unlistenable for me. Nora Jones sounded like the worst singer in the world with that amp, any time her voice reached up it sound like she had a horrible scratchy, distorted voice. This sound only revealed itself when the amp was installed. I replaced the amp and everything went away. I once had the same issue with a DAC and another time with a CDP (but with the CDP it was only at higher volumes where it became apparent - and much less so than with the Dreadnaught).

Your steps seems to be in good order, but try to find people in your area before you start replacing components on a guessing basis. You may replace half your components before you find the real culprit if it really is even component related???

Its probably a combined effect of everything together more than due to any one thing.

I'd get some decent ICs as an easy and cheap thing to try as a first step.

I'd recommend a used pair of MIT Terminator 2 or 4 ICs for < $50 used as a first step to help tame things down a bit and get more out of what you have before changing anything else.

Next I'd look at some decent but not expensive speaker wires also for an upgrade. DNM Reson would be my first choice.

From there, perhaps a tube DAC upgrade to use with the CD player or a tube pre-amp if the harshness exists in multiple sources, not just digital.

For a tube DAC to help tame things further, I'd recommend an mhdt Paradisea or even better a mhdt Havana if you can squeeze it.

Is there any adjustments on the pre-amp or amp that might be turned up too much. Isn't it possible to have the inputs too sensitive?