I think I've figured it out!
Safari hanged today, so I fired up Console.app again and started streaming. This time, I noticed quite a few Time Machine errors. Failing to mount the disk, failing to find the resource, etc. I checked the menu bar, and sure enough, Time Machine said "Looking for backup disk ..."
I let it continue without intervention.
Eventually, it found the disk. As soon as it changed from "Looking for backup disk ..." to "Connecting to backup disk ...", Safari loaded the page.
My Time Machine is via a Time Capsule (last generation). Without verifying, I hypothesize the HDD had gone to sleep, and it took a minute to spin it up and connect, during which time Safari stopped working. By any chance, do you have a similar setup?