How do non-LaTeX users handle citations?
It isn't exactly hard to do references by hand, in any word processing system, just boring and a bit fiddly at times. However, given the number of questions here on references it isn't like the citation managers aren't fiddly as well.

Having in the deep dark past written citation handling/numbering preprocessers/code for nroff, spinoff, and straight TeX (since LaTeX was too foofy for real scientists), those aren't that hard to do either with a modicum of programming skill. After the pain of writing the system in the first place, it is all gravy from there. 

Even 30 years ago programs such as EndNote made it pretty easy to get a citation manager that integrated well with Word (plus or minus various perversions caused by Word itself). Nowadays there are more options, and it mostly comes down to finding a word processing system and citation manager that work together and that you will stick with for long enough to make it worthwhile. 

That last part is key - after investing the time and energy to build up and learn a system that works for you, it is a pain to change. All of a sudden you move from grad school to a post-doc, and your advisor doesn't use the same tool set -- what now? Do you switch from LaTeX to Word (and the citation manager as well)? Or do you do all editing/revisions on paper? Once you have switched (for whatever reason) a few times, one might decide that doing it all by hand really isn't so bad.