Is there any way to get starman to put timestamps in its error log? It would be very useful to be able to tell whether an error just happened or if it's three days old.

You can run your starman script under daemontools as a service and get the log with multilog, which prepends a hi-res timestamp to each line. (https://cr.yp.to/daemontools.html)

(In fact, you can probably use multilog to capture the output of any program, see https://cr.yp.to/daemontools/multilog.html.)