Was there some particular design theory or constraint that made a 36 bit word size attractive for early computers?
Beside integer arithmetic, 36 bit words work quite fine with two different byte sizes: Six and nine. Six bit was whats needed to store characters of the standard code for data transmission at that time: Baudot code or more exact ITA2.
As opposed to the various power-of-2 word sizes?
There is no inherent benefit of power of two word sizes. Any number can do.
Even more, there were no 'various power-of-two sizes' in the early and not so early days. Before the IBM/360 settled for a 32 Bit word size and four 8 bit bytes within a word and two nibble in a byte, power-of-two word sizes where an extreme exception (can't come up with any beside SAGE and IBM Stretch). The vast majority used word sizes dividable by 3 not at least to allow the use of octal representation. Before the IBM /360 with its 8 bit bytes, octal was as common to computer scientists as hex today - heck, Unix carries this legacy until today, making everyone learn octal at a time when hex is the general accepted way to display binary data.
Now, the reason why Amdahl did choose 8 bit bytes is rather simple: it was the most efficient way to store two BCD digits within a bye and thus a word as well. Operating in BCD was one main requirement for the /360 design, as it was meant to not only be compatible to, but as well replace all prior decimal machinery.
Is what seams today as 'natural' use of power of two it just a side effect from being able to handle decimal by a binary computer.
Conclusion: As so often in computing the answer is IBM /360 and the rest is history :)