I think this is a good idea. I prefer either videos, GIFS or a well documented screenshot tutorial. But I think this should only be a complement to aguide not something that should replace normal instructions.
Example:
to update your computer simply type:
sudo apt-get update && sudo apt-get dist-upgrade -y
GIF of the procedure. This is here to improve the feedback to the user so the user knows what should happen and when something has gone wrong. If a guide is old then that will most likely be shown in the GIF.
But what about slower loading of the page if a GIF is there?
This can be solved with a click to load GIF button on the GIF. Yes it will be slower than just text but a guide with screenshots will also be slower than just text. And I think GIF's are a better way to show the procedure than screenshots.
But these are just my ideas on the topic, I can't speak if it can be done on the site/network or if they are willing to do it. I like it though.
My solution is to use GIF's instead of videos, why?
Gif's are simpler and cleaner, in a video users may add a hello statement and a long speech. But a GIF is simple and a better solution to the problem. The problem as I see it is that a normal tutorial may not tell the whole story, if it only contains type these commands in a terminal and it will work. If you have a GIF that does that then the user knows what to expect and they know when it goes wrong.
Videos have many problems with being heavy for the network and storage, which GIF's aren't.