The amp now sounds overly bright to my ears with piercing highs. So I clipped the 0.001 cap on the first voltage divider after gainpot 1. The amp sounds less bright, but not as articulate as with the cap in, which made me decide to put the cap back in.
My question now is what is the best way to tame the brightness a bit
I would remove all the cathode bypass caps on the first 3 gain stages and just focus on one (maybe the first, because that's traditionally the one that produces the max gain and overdrives V2), I'd start with a big cap, like a 25uf to make it simple, then see if you can fiddle with your tone-stack to get whatever range of highs, lows, and mids you want the amp to be capable of. If you take low freq. out of the signal before the tone-stack, you can put them back in at that point.
On your tone-stack, I would change the way you've wired the 500K pot to be a variable resistor between those two junctions (like you commonly see with vox,marshall,fender,etc).
Also, isn't the cathode bypass cap 10uf on the output stage quite a low value?
As for the ECC83 vs ECC81: survey all the great tube amp schematics, you'll never find a ECC81 at V1. the ECC81 is a good work horse tube (LTPI,reverb driver,etc) but isn't anything special when it comes to preamp gain duty. All you've done is lower the overall gain exiting each stage as it hits the grid of the next. I would stick your best ECC83 back in V1 and change your voltage dividers to render the same +dB at each stage that you have with the ECC81.
It might be easier to tune if you change your first couple of preamp voltage dividers to what is pictured below right. If you make R1 & P1 = 500K you'll get the same end result as what you have at left. When pot is at "10" A->B is 50%, When pot is at "5" A->B is 25%. There is a change to low freq. roll off from the CR filter, but you move the cut off from something like 30Hz to 10Hz... so no big deal. It might mess with your .001||220K, but something like a 330pf||500K might give you same thing.

the advantage if this is you can stick some 1M or 2M pots in there as
"R1" trim pots, and then tuning the amp can be done while the amp is powered on, and you don't have solder/desolder, etc..
Stick a ECC83 back in there, fiddle with the trimmer
"R1"s (in that R1/P1 arrangement) and don't be worried if the second stage has a 1M on top of a 500K pot. If that's what sounds best!