Well no, rclone doesn't necessarily need to run on each server.
As long as one machine has access to the other data (via network shares or anything else you can configure as a remote), it can pull that data in and send it where you specify. As long as you don't regularly need to send large data between two off-site servers then having a single rclone remote that does all the rclone work for the entire local network would usually be the most sensible. You'd go a little mad trying to remote control a 100 different computers
If it makes it any easier to think about, an rclone remote is no different from running rclone normally. The difference is that you can sit in your office or at home on a bad wifi connection and tell the server "hey transfer these 300GB of data to that other server" and your server will do that for you. If you ran rclone locally then all the data would go to your machine before it reached it's destination. But when there are many users this also means you can share any caching, have to worry much less about concurrency-problems and much else, so there are many add-on benefits. Imagine using the RC as the equivalent of every user running up to the server-room to use rclone directly from the server and that's the gist of it.
I can't read your admin's mind and know exactly why he thought it was critical you do it this way - but it's probably related to the nitty-gritty details of the configuration he set up that may not work optimally (or in the worst case even produce errors and corruption) if you don't use the RC for the commands. It could be set up either way, but for a large setup the RC is the obvious way to do it properly and efficiently, so I'm not at all surprised about that choice.
--rc and rcd both start up the rclone remote-control server that sits and listens for commands.
rcd does this, and only this
--rc is more like "and also do this in addition to the main command". It's a flag so you can't run it alone. It's just an option to another command. I use it on my mount command for example so that I both get the mount but also have a RC server for that mount that I can send commands though if needed.