AMD has MSAA as well. Some games have options for it on both cards as well. It's nice but it has limits, and it doesn't work in every game since it's still a rendered form of antialiasing.
I'll explain it in a nutshell, since it looks like you're still uncertain what jaggies really are and how they work.
This is going to start out really basic, so I'm sorry if I inadvertently offend you.
Screen resolution is measured in pixels of course. 1920x1080 is a measure of pixels. 1920 pixels wide and 1080 pixels high. Sorry if you already know that. In games, jaggies are the natural effect of pixels making a shape. Imagine building something out of Legos. Since they're blocky, whatever you build is going to have jagged edges (jaggies). If you make the Legos smaller, the jaggies will be smaller. Legos are like pixels. Pixels are blocks used to make a shape, so the edges will always be jagged, but by making the pixels really small and adding a lot of them in a high resolution, we can make the jaggies so small that they become hard to notice. The jaggies you're noticing are simply the normal result of making a scene out of pixels. They show up in every game, on every screen, on every console and PC, though some developers do a pretty good job hiding them. The Xbox 360, for example, had a full core dedicated to running 2xQ, a form of antialiasing used in the majority of 360 games. The Xbox One cut out this feature for unknown reasons (possibly to save money.)
Now, antialiasing is intelligent blur. It hides jaggies by fading out the edges of shapes over several pixels, so that jaggies become too faint to see. I'm oversimplifying it, but that's basically what it is. Some antialiasing is less precise, like 2xQ, FXAA or SMAA, so you can actually notice the blur if you look closely. Some AA is more precise, like MSAA or SSAA, so you can practically never notice the blur. However, antialiasing does not work in every game. The simpler forms (2xQ, FXAA, SMAA, MLAA, AAA) work in a lot more games than the more accurate kinds of antialiasing.
Since jaggies are just the result of making shapes out of pixels, the screen's colors matter a lot. Nicer screens with better colors tend to hide jaggies better. Poor screens often exaggerate their contrast or apply sharpen filters to "enhance" their appearance, which makes jaggies extremely noticeable. Some TVs also include edge-smoothing methods to automatically reduce jaggies.
Taking your PC or X1 to a repair shop won't help, because jaggies are unfortunately not a defect.
It's all sort of a cluster**** and there's no perfect way to get rid of jaggies completely, but there are a lot of very good ways to get rid of 90% of them.
Do you mind taking a screenshot of the antialiasing section of your CCC? You never mentioned whether you reset it, so before anything else can be suggested we really need to know how that's set.
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