Rereading your original message, I'm guessing that upgrading to a new Shiny Server version will eliminate this "Failed to render error 500" in your log, but not resolve whatever the problem is with your app. The error message is vague on purpose, on the off chance that your error messages happen to contain sensitive information like usernames. If you'd prefer helpful error messages, you can turn off the vagueness by adding the line sanitize_errors false; to the top level of your /etc/shiny-server/shiny-server.conf and reloading or restarting the shiny-server service.
Whether errors are being sanitized or not, you should be able to find log files in /var/log/shiny-server/ (note that, confusingly, this is not the same as /var/log/shiny-server.log). The /var/log/shiny-server.log file is what you've posted, and it contains the output from the shiny-server process itself. The errors that are causing the unhelpful messages in the browser are originating from the R processes, and those are in the /var/log/shiny-server/ directory.
One last complication--sorry. In an effort to not fill up your hard drive with spurious log data, by default Shiny Server deletes the log files in /var/log/shiny-server/ if the R process that generated it exits gracefully. If you're not finding the log files you're looking for, you can turn off this automatic deletion by adding the line preserve_logs true; to your shiny-server.conf, and again, reloading or restarting.
The most common source of errors that only appear on the server are 1) some package that your app uses is installed locally but not on the server (at least not in a place that the shiny user can get to it), or 2) you've got a hard-coded path in your app logic that points to a location that doesn't exist on the server (use a relative path instead).