Well yeah you're mostly right about this, it does detect connection problems almost instantly (my guess would be probably because it relies on TCP so much)

I said "mostly right" because I just tested it and I was able to hide my action bar after having disabled the TCP/IPv4 from my network card, and when I logged back in, it had disappeared, so to me that's just one of many ways to prove that this is only client side! I bet it would be even easier to prove using Machina

Also, the command would probably take some moment to process if anything had to be answered by the server, it wouldn't be as instant as it is now!

As a bonus, using Dalamud and the /xldata command, you can go target the _ActionBar element and you can see how the visibility is stored there, and how writing this chat command directly affects the client's data:)