Just a heads up, you have Teac spelled "Teah." Also it's "The World's Best DSD DAC Under $1,000" and has the "Celebrated BurrBrown PCM1795" and "3-dimensional" MUSES8920 op-amps. So obviously you are totes wrong.

Considering how blah entry level Esoterics are, I am completely unsurprised that a Teac DAC would be crap.

The reviews have a very interesting list of qualities being evaluated, but you are missing Timbre.   When you have " II. Tonal Balance " that is the "Macro" and Timbre is the "Micro" (just as you have Macro-Dynamics and Micro-Dynamics).

So, the Balance of Harmonics needs to be a separately evaluated category.

Are there any measurements that can show or maybe even hint at the difference among DACs' timbre?

creating an accurate ladder chip (without trickery) is nigh-on impossible to do right because of the tolerances involved.
For 16 bits it is already physically almost impossible (hence the ENOB being smaller than 16 bit).
a 24 bit pure ladder chip is not possible with 1 R2R ladder, some trickery is involved.

The fact that a chip accepts 20, 24 or even 32 bit data and is called a 24 bit or 32 bit DAC chip doesn't mean it actually reaches that physical resolution.
In fact 32 bit is impossible to resolve in an analog way as the S/N ratio is already factors higher.
I would be really surprised if an ENOB of 24 bits will ever be created.

I would expect the production process for ladder chips that are really high spec'd to be very costly and probably during a selection process based on minimal requirements they may have to ditch a lot of chips addong to the costs.
Accuracy of these chips is shown by the suffix b.t.w. I expect the -K suffix one will be the most expensive but at RS all the different suffix chips have the same price. Why bother buying /offering anything less than the -K suffix chips ?

1 bit chips are much easier/cheaper to create and can have a good ENOB.

As long as there is a demand manufacturers will supply... they only care about sales or prestige.

Subsequently you can ask if there are any headphones that can reproduce those frequencies and the next question would be if there are any recordings made that have useful information and lastly ... would one be able to hear that and would it have anything to do with 'timbre'.