Server packages on Steam are handled 2 different ways (well 3, you can combine both)):
- Either the package is free to anyone, meaning it is available to "anonymous" login
- Or the package is manually, by the developer, granted to base game owner (logged in account) and is therefore not available to other logged in accounts not owning the base package.
- An outlier case exist when the package is granted to all logged in users, this is something ive personally not seen much as "anonymous" login exist.

I also recommend to just use SteamCMD with anonymous login to maintain your servers as you can script it easily (commandline arg "-runscript scriptName.txt" for SteamCMD.exe), it will not have Steam service running so it will not interfere with server in any way (running multiple of the same game is not possible while Steam is running, this also applies to servers).