You bring up a very interesting subject ...
First of all, Linux clients do exist for Microsoft's "Remote Desktop" protocols, and all of them work very well. But,
Linux does things differently.
Linux actually uses a
client/server strategy for windowing. This is called
XWindows (the name is coincidental), or lately,
XOrg. On a single computer, both the client and the server are running at the same time.
(Betcha never knew that until now ...)
This strategy was inherited from Unix®, and it is actually many years old now. (Macintoshes, although built on Unix, provide a client application but do not use it all the time: if you run an app that needs it, it will silently and automatically start, and run as long as it is needed.)
"The server tells the client what to draw, and the client draws it." This means that you can remotely connect to a "headless" server that doesn't even
have a graphic card, then conduct a full GUI session with it, and do so very efficiently. The only requirement is that the server is running the "server" process, and that you are running the "client" process. The compressed data stream which passes between the computer consists of mouse/keyboard events and drawing commands.
Microsoft Windows® implemented things in an entirely different way, so their "remote desktop" protocol – even though it works well enough – is noticeably less efficient when used over a busy network. Remote Desktop has to futz around with bitmaps: very clumsy by comparison.