I'm putting this together in the hopes that it helps someone else who has run into / may run into this issue. It's possible that this is documented in the Install Guide linked below but I did not see it there and, when I looked again today couldn't find anything on this topic.
The Tableau Server on Linux: Everybody's Install Guide is a fantastic resource to get Tableau running on Linux and, after following the guide, our server has been running smoothly. Today I wanted to check on some traffic numbers for a specific view and, when I tried to use the "Who Has Seen This View?" function, I was met by this error message:
"Oh no, the server has failed catastrophically and cannot connect to Postgres. There goes a full day rebuilding and restoring the deploy!" was my first thought. I started loading vizzes and everything seemed to be completely fine otherwise ("whew, well at least it's not that bad"). I went to the Traffic to Views dashboard under Site Status and got a similar error:
I logged into my Server instance (we are running on Amazon Linux 2, FYI) to check what postgres packages, if any, were installed. As it turned out, the errors were correct, there were no postgres packages installed at all (which seems a little weird since Tableau uses PostgreSQL heavily). So at least I was comfortable that there was no failure and that the errors were reporting accurate information.
When I went searching for the appropriate drivers, I came across the actual answer to the question I unknowingly set out to answer. I went to https://www.tableau.com/support/drivers and searched for Postgres drivers for Linux. On the PostgreSQL driver page for Tableau Server 10.5 - 2018.3.1, this is noted:
Important: Built-in administrative views will not display unless you install the PostgreSQL driver on every node running the VizQL Server process.
So, it looks like there is a currently undocumented step necessary to complete the installation on Linux. Everything will work as expected without this driver aside from the Admin views (at least as far as I've seen, it's possible there are additional downstream impacts that I haven't run into yet). It would be great if this driver were installed as a dependency of the .rpm file used to install Tableau Server but, for the time being, just grab the Postgres driver, install, and you should be all set.