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 what's needed to store characters of the standard code for data transmission at that time: Baudot code or more exactly 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 were an extreme exception (can't come up with any beside SAGE and IBM Stretch). The vast majority used word sizes divisable by 3, not 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 generally accepted way to display binary data.
Now, the reason why Amdahl did choose 8 bit bytes is rather simple: A byte size chosen had to be at least 6 bit to store a character, eventually 7 for the upcoming ASCII, but 8 would give the ability to store two BCD digits within. Any larger byte size would again waste storage with this important element. 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.
What seems today as 'natural' use of power of two is 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 :)
