Screen Saver and Screen Blanking don't happen

Do you have a wireless mouse and/or keyboard? Could they be active somehow to keep interacting with the computer? The Apple help section on this says "choose how long your Mac needs to be inactive before the screen saver starts." If it is never inactive, the screen saver won't start. Have you tried the "sleep" command in the Apple menu, does it sleep then? Have you tried using "hot corners" to immediately activate screen saver, does that work? Also, look under the Bluetooth menu and remove any devices you are not currently using. Something else may be pinging it via Bluetooth.

Try disabling the server function completely for a few minutes to see if normal behavior returns.

I thought only keyboard presses, mouse activity, or screen activity would trigger the screen.

Those things wake it up after it is sleeping or in screen saver, but yours is never going to those modes in the first place. Your screen shot shows both sharing and bluetooth and either or both of those could be preventing sleep (or screen saver).

It seems that you may be sharing something - internet? files? screen?

and that may be what is preventing sleep.

Go to System Preferences->Sharing and see what is there. Uncheck any services that you do not need.

If you are using it as a server, other activities on other devices will impact the server, so it's hard to say why this happens without knowing what might have changed on those other devices.

Maybe also check your energy saver settings regarding "wake for network access" thought which network activity could impact the server and its ability to sleep. What has changed on your network? Something as simple as a minor update to the MacOS or iOS on any of the devices could have changed how they interact with servers.

When you look at Bluetooth, are there other devices connected or trying to connect, other than the mouse or keyboard you normally use? Are you connected to a hub or dock?

Spotlight indexing normally happens with the screen in saver mode (or with the screen off) with no impact on the screen.

Do you have any network monitoring tools, Little Snitch, third party firewalls or any antivirus or security tools? Anything at all that updates itself in the background? (Lots of software does that, by the way.)

If the answer is no to everything, I would try an experiment to troubleshoot. Disconnect from WiFi and from the internet and everything else except the mouse and keyboard (replace them with wired ones if you have them, in which case you can turn off Bluetooth as well). This test need only last a few minutes, then you can go back to normal connections. If it still won't go into screen saver properly, then something is installed most likely causing this. In which case, download and run Etrecheck and post the output (it is anonymous) here with the Additional Text button below.