You can do both. The default is Big Picture mode, but you can back out if it and get a desktop.

But, you’ll basically have to be a foot or two from the PC to read anything unless you have a desktop environment set up for a large screen (KDE plasma has a TV version)

You can control the computer but it boots in big picture so you need to escape it to get to the desktop

Can control it in general, but the first few times you’ll have to run over and look at the monitor to do stuff.
It generally assumes it’s being used “close” to the computer, so instead of complex pairing, it just shows a code on one you type into the other.
Sometimes windows will get antsy and pop up a dialogue that can only be interacted with locally, but it’s only one or twice ever.