Regular restarts, like, weekly? Really? That doesn't sound robust to me and particularly not good when it's executing extracts 24x7 as well as providing subscriptions (built-in and VizAlerts). We have users from all the Americas (western hemisphere) and some users in the rest of the world (mostly UK & Europe). Bringing down our Tableau Server "for good measure" seems like crazy talk
What is to gain by restarts?
As for the backups & ziplogs, I agree, the more often the better (depending on one's BI environment). We do ours nightly with the a running backup (no stopping the Tableau Server) and keep a rolling 5 days worth.
Everyone has their "UpTime" requirements and every deployment is different (just like snowflakes).
I see customers do each of the options.
Tableau Server robustness would be a great topic to collect data on and I would love to do that if I had the time and access to all logs. It would be interesting to see if there are correlations between types of deployments (HA, x number of nodes, whatever and everything). And if so what those correlations are and how they are most effectively managed.
In the real world, I see a lot of customer deployments that have issues that resolve immediately when the server is restarted. Sometimes they can even restart one node and get lucky. Somepeople never restart and do fine (although I'll bet their log files get huge).
As a general rule, restarting Tableau Server on a regular basis and cleaning it up while it is stopped is a good practice to keep the deployment on a steady track.