In asking the question of why one might want to use 31-ET, one might ask: "Why not some other tuning?" 34-ET is one such tuning that is often considered, as it has a similar width to its divisions of the octave, and offers a better match to the fourths and fifths, as well as minor thirds.
While 34-ET is a rich and interesting tuning, providing certain advantages over 31-ET, there are several reasons that it tends to be more difficult to learn, sing, play, and understand, than 31-ET. These reasons include:
There are only 14 intervals (7 and their inversions) of the 34-step scale in 34-ET that are in-tune and usable in the sense that they are in-tune and can be put in harmonic context.
The overall effect of writing music in 34-ET is that a composer is forced into writing tightly-controlled diatonic music, being careful that intervals line up properly so as not to fall outside the few useful intervals of 34-ET. If the composer ventures outside of this box, there is an abrupt cutoff from the familiar intervals of Western diatonic scales, into a strongly xenharmonic realm of strange and dissonant intervals that, to most ears, even trained ones, will sound highly out-of-tune. Furthermore, the music must be notated so as to distinguish between major and minor whole tones, leading to cumbersome notation.
Perhaps there is potential for some creative composition using the contrast of a closely in-tune diatonic scale with a more discordant, xenharmonic realm of intermediate intervals, but I as a composer have found this system too hard to work with, and I strongly prefer 31-ET for the reasons described here.
Although I'm obviously an advocate of 31-ET, due to its simplicity, I think there are better tunings than 34-ET for composers who want some of the key features of 34-ET. In particular, 34-ET distinguishes between the major and minor whole tones, which might be a desirable feature. The next smallest equal temperament after 34 that is usable overall and that also does this is 41 equal temperament.
41-ET seems to combine the strengths of both 31-ET and 34-ET. Unlike 34-ET, 41-ET provides a good match for the intervals involving the 7th and 11th harmonics. It also addresses one shortcoming of 31-ET: it distinguishes between the undecimal major third (14:11) and the septimal major third (9:7).