Well, in a few turns, I'm in a slightly different (hopefully better) position.
I sold the SR to another BF member after trying out a few tire sizes and deciding I would not have much success in finding a wider tire without modding the frame, something I'm not looking to do.
Then I bought this 1980 Univega Specialissima. It seems to be in great condition, though I need to make a few adjustments to the cockpit. That same BF member who bought the SR from me offered to rewrap the handlebars for me, as well as swap in a Nitto Noodle bar, so I'll grab a few new pics when that happens, but in the meantime, here it is on day of purchase:
I'll note that the ride quality was remarkably smooth in comparison to what I was expecting. Perhaps my expectations were too low given all the Rivendell stuff I read, but I was pleasantly surprised. I think this will be a more viable rando candidate. The only thing that gives me pause is top tube length, but we'll see how a stem swap can help that.