Does one time pad randomness help brute force attacks?
There is no such thing as a random key.  There are only randomly generated keys.

What I mean by that is that randomness is not a property of the key (or message, or number, or whatever), but of the process that generates it.  For example, it is not meaningful to ask whether the number 5 is random, or whether it is somehow more or less random than the number 4.  However, if I pick up a die and roll it, and it lands with the "5" face up, then I can meaningfully say that I have just randomly selected the number 5 (and we can also meaningfully discuss just how random or not the way I rolled the die may have been).

In English we often do use "random" as an adjective to describe a thing (as opposed to an action), when we really should be saying "randomly chosen" or something similar.  But this needs to be understood as just a linguistic shorthand.  "A random number" really means "a randomly chosen number", and that's all there is, or can be, to it.

Why am I stressing this point so much?  It's because when you ask something like:


  If key TQURI is "measured" as less random than XMCKL, ...


you're revealing a fundamental confusion about what "a random key" really means.  As I wrote above, randomness is not a property of the key string itself, so there's no way to assert that, say, the string "TQURI" is somehow more or less random than "XMCKL" or, for that matter, "ABCDE".  Even trying to make the comparison is meaningless.

All that we can meaningfully ask is how likely it is that each of those key strings would appear if we, say, randomly picked five letters from a bag containing each letter of the alphabet and wrote them down (returning each picked letter to the bag before picking the next one).  And the answer to that question is that each of those strings (yes, even "ABCDE") is equally likely to be the result of that process.

Thus, if our one-time pad was generated using the method I described (or by any other equally random means), then none of those proposed keys is any more likely to be the correct one than any other, and so examining them yields no information about the message.



Ps. Yes, this also means that randomness testing is, strictly speaking, a fool's endeavor: a truly random process is equally likely to output any string of a given length, so no such string can be said to be any more or less random than any other in that sense.

The best we can do is to look for distinctive features that may appear disproportionately more often in the results of certain kinds of non-random (or incompletely random) processes, and declare the observation of such features as evidence that the process that generated the output may not be perfectly random.  In that sense, it would be more appropriate to call any such tests "non-randomness testing".

Even then, in doing so, we must necessarily accept the possibility of "false positives", i.e. the possibility that even a perfectly random process can, and will, occasionally generate outputs that look like the output of some non-random process.  A perfectly random 5-letter string generator must, necessarily, generate the string "AAAAA" with probability $1/26^5 \approx 0.00000008$ &mdash; and if it did not, it would not be perfectly random.