0x00A9

Ok so what is the issue you are facing?
Players cannot use Skins? -> Steam Rate limits you, every skin that loads; is downloaded by Steam itself. They limit you after a certain amt of time.
Server not loading Skins? -> Steam issue again, after downloading so many skins, steam will rate limit. 

I suspect the problem is network-related.  
And you would be correct. additionally you said I ran Steam there, the workshop page didn't open, nor did the store, this is indeed Steam saying 'hey you downloaded a lot of workshop content recently, you have exceeded a temporary limit'
Limits are usually : 5min, 30min, 1 hour. 
Steam does not kick/ban you for downloading content either.

Also, any dedicated Server on steam should contain \steamapps , thats how Steams dedicated servers are, not just rust alone, so your server could also be corrupt.

My server has SteamApps, but there's nothing there. Server updates are downloaded there, and the downloading, temp, and manifest folders are there. For many years, this folder has never weighed more than 5 KB, even when skins were working.

name

My server has SteamApps, but there's nothing there. Server updates are downloaded there, and the downloading, temp, and manifest folders are there. For many years, this folder has never weighed more than 5 KB, even when skins were working.

You can host workshop content to players using your server (your-rust-server-name\steamapps\workshop\content\252490)- which would effect you and others on your server.
You can have workshop content in your own steamapps dir, (C:\Program Files (x86)\Steam\steamapps\workshop\content\252490)