This is not an issue of the clients not getting the updated library while they are connected, although I note that was not clear from the original post.
Media Server is running 24/7, clients connect and disconnect at various times but all seem only to be served the library at the time Media Server first started running. As noted in the initial post, the same behavior applies when I just have the clients read the library directly from the network share where it is stored instead of via Library Server.
However, when I access MC on the server itself, I see the library with all updates since it was started. But in order to get the client PCs to be able to use/see this updated library, I have to restart MC on the server.
MC on the Server should not running in read only mode as no alert dialog comes up when I run it and no other PCs have the library open. Also, if that were true when I close and reopen MC on the server, (i) the library shouldn't be showing the new changes to the library, those should have disappeared when I closed MC running on the server, and (ii) it should not be automatically downloading podcasts.
Not knowing how MC is architected makes it hard to guess what the problem might be, but it is as if (a) the MC on the server is keeping changes from downloaded podcasts in memory and not writing them out to disk, while (b) Library server is reading the older library from disk rather than the version that the server MC has in memory.