A load balancer is usually a network device used to transparently direct and distribute connections among a pool of servers. When one of them gets into a bad state, the users who happened to get connected to it experience issues while those connected to others do not. Restarting the web browser, or starting a new one, creates a new connection or session between the browser and the load balancer. It's my guess from this issue and from observing the patterns of my sessions getting logged out over the past few months that the hosting provider is using load balancers and that they are, at least sometimes, not entirely reliable.
Ironically after I made the original post there was an ad for "used F5 load balancers" at the bottom of the screen.