I agree that while twitter can be nice and fun for writers, and I enjoy talking to writers on it, it is certainly not a requirement. And if you don't like it or it is a time-suck, WRITING is way more important! But I wanted to speak to this:
Twitter isn't necessary for the querying process. But you can learn about an agent's preferences and the like. I am also a fan of Twitter for getting a read of an agent's personality. I have seen some of them tweet mess that caused a deep side eye and I cut them from my list.
People always say writers should watch themselves on Twitter. The same holds true for agents because WE are also watching.
I tweet about all kinds of things, from whatever book I'm reading, to what my dog is doing, to travel pics, to recipes, to some random funny thing I saw. And plenty of dumb jokes, silly banter etc in between.
If somebody disliked, say, my opinion about Bates on Downton Abbey, or pics of my dog, enough to want to not query me, then good. We would not be a match. (Because obviously Bates is the worst and my dog is the best! ;-)
I have really only had one or two people freak out on me on twitter. Once I tweeted that book piracy is wrong, you need to support the business you want to be a part of, and that if you claim you want to be a writer but you simultaneously steal books from other writers, you need to reexamine your priorities. Some person started a fake account just to call me names. LOL. oy.
Another person flipped out on me for "engaging in trivialities" ("height of mundanity!" I believe she called it) and said she "didn't know how my clients could stand it!" and she huffily unfollowed me -- funnily enough, her manuscript was under consideration by me at the time. So, you know. I rejected it, because obviously we are not a good fit.
Personally I LIKE trivialities on twitter. I go on there to have fun and have a laugh, talk to my peers, catch up with old friends and make new as well. I live in the country and work from home - for me it's basically the equiv of an office water cooler.
As you say, so much of working well with somebody does come down to a personality thing. If twitter helps you decide who not to query, that's a GOOD thing. So instead of watching themselves more... maybe agents should watch themselves a little LESS! ;-)