GStreamer is a library which handles audio and video processing. It's usually used by GNOME applications and their derivatives for multimedia decoding and encoding, usually the former to display / play video and audio files.
WebKit, an HTML "rendering engine" (core of a Web browser), is among the users of GStreamer; multimedia files abound on the modern Web, and thus WebKit needs to support multimedia files in order to be competitive in the modern Web rendering engine arena. This last point matters because the Ubuntu MATE Welcome application (and even the Software Boutique, I think [!]) uses WebKit to display the fancy user interface of the Welcome application. (Why WebKit is used for the purpose remains a mystery to me. My best guess is that the designer of the Welcome application wasn't so comfortable writing a conventional program and adding all kinds of special effects and animations to the program code, and so decided to write the application mostly in HTML.)
GStreamer is highly modular; by itself, GStreamer can't do very much of use. You therefore need to add plugins to GStreamer to extend its functionality, to play MP3s, for instance, or use special hardware features to decode and encode multimedia at super speeds.
The latter describes the VA-API plugin that you've been directed to remove; all that you'd remove is the ability for GStreamer to use special features of modern hardware (usually graphics cards) to turn audio and video encoding / decoding from a somewhat slow speed into a quick, real-time process that uses relatively little electricity. You would not remove the entirety of GStreamer by removing this gstreamer1.0-vaapi package; if you did remove the entirety of GStreamer, you'd probably remove half your desktop applications. (GStreamer is needed by WebKit; as well as being needed by the Welcome app, WebKit is needed by Zenity, which is used by the MATE window manager Marco. Many other intricate tangles of proverbial cyber threads attach GStreamer to dozens of other applications, and all in all, you'd have a very crippled desktop.) Fortunately, this workaround does not involve uninstalling all of GStreamer.
Now you're probably saying, "Less electricity use? Isn't this a pretty important package then?" But the fact is that really only computer systems from the last 6 or 7 years have this special hardware required for VA-API support. Furthermore, if you're running Ubuntu MATE inside a virtual machine, chances are your virtual machine lacks any such (virtualized or real) en/decoding hardware! Unless you actually have a system which can truly support VA-API, removing the package will not cause any harm.
As for why this plugin interats with X.org so bad: I looked at the pertinent bug reports, and it's fairly clear to me that nobody really knows why this VA-API plugin for GStreamer crashes X sometimes. Some say it's a bug in the VA-API driver itself; others note that it must be a bug in X since the problem occasionally still persists even after the package is removed.
My personal opinion, as a guy who really doesn't know what he's talking about in this exact regard, is that on some hardware configurations, the same hardware circuits in charge of hardware multimedia en/decoding are the same as the ones used by graphics acceleration (for displaying stuff on-screen really fast, as opposed to simply decoding video really fast, leaving the actual display process up to another component of the system). Or it could be that it's not the hardware at fault, but the software: I've noticed that under a lot of circumstances, the driver for VA-API comes from the Mesa project, which is primarily focused on hardware-accelerated graphics rendering. Maybe, just maybe, some of the drivers for VA-API on certain hardware somehow conflict with the drivers for graphics acceleration.
But that's just a W.A.G on my part, and I really have no business preaching stuff that's outside my field of expertise. So I'll leave facts at facts and move on. Still, I hope my diatribe was at least half-useful for clarification purposes.
) I wonder if there is any shot that the root cause is the same for both . . .