Unless you have lost the ability to use language to begin with. Back in the 1960s, the audiences might not have been all that familiar with such a possibility. Today, hospital porn has made selective brain trauma mainstream.
People who have suffered damage to the output parts of language processing might still express themselves, though. Say, how about giving Pike a pen? Even if he couldn't write language, he might draw pictures. (In the worst case scenario, just tape the pen to the corner of the chair. In a more realistic one, hook it up to a mechanical arm Pike can control, the same way he might control a voice box but doesn't because he can't form phrases.)
We meet Pike "months" after his big accident. Perhaps the boxy whole-body prosthetic is only the first step in his partial recovery, and arms and legs will be added later? Spock makes his own artificial hurry there.
But yes, setting up 20 questions ought to be informative if Pike truly and coherently can express the concepts of "agreement" and "disagreement". And no doubt Pike often got to play that game. But his bout of yelling "nonononono!" involved him refusing to cooperate - he "almost agitated himself into a coma" when "questioned". So yes, these geniuses did ask.
In contrast, giving the UT a free hand at interpreting the yes'es and no's would not be helpful. Somebody would then have to interpret the UT, resulting in messages no more coherent than those from the oracle of Delphi.
Timo Saloniemi