Thank you for the clarification.
I'm using the server operated by Remote Utilities, not one running on my own machine, so I don't think I can see its logs - or can I?
Yes, sorry for misleading, I misunderstood this part of your message, thought that you've meant the self-hosted server. You are right, in order to check the Host logs you need to be able to connect to the remote Host machine.
Could you please also let me know if you mean something that would show when the Host was online for the last time? If this is the case, then we plan to implement such a feature: a column in the Viewer that will show when the Host was online for the last time, so it would help to troubleshoot some connection issues like the one you've described above. We already have this feature in our wishlist, but, unfortunately, cannot provide any specific ETA on it.
Let me know if you have more questions.