This is the thing where I don't understand YouTube. This is not on the net where somebody is making money off it, so why is it blocked? The NBC website has the Lipinski/Weir commentary.
Depends on what they're doing. And yes, companies make money off of it.
If they're directly removing the video, then I'm not familiar with that, so don't have much to say about that. If they're doing it via Youtube's matched third party content, then it's so they can put ads on your videos, making money off of it. There's an automated matching service that Youtube provides (basically will scan every upload against the client's content), so Youtube makes money off of companies who pay for that service. So basically, companies can make money by putting ads in your videos, and Youtube makes money from companies who pay for the automated matching.
Basically with matched third party content, the company claims rights over your video if you leave it up, hence they get to put ads on it (or take it down if they want). This is even if their "right" is due to something tangential like sounds coming from a background TV. The problem is Youtube's automated matching is extremely imprecise; there's been reports of it matching things like bird calls and stuff. And it's up to the uploader (not the company) to demonstrate why the company should not have rights over the video, rather than the way around, which is how we usually think the legal system works. The system is basically like companies taking over your house until you can prove to them that you have a valid deed on the property; or civil forfeiture in the United States.
I've gotten the matched third party content notice for my piano videos, 5 from Chopin's Fantaisie Impromptu alone. This even though Chopin's been dead for a long time (so his works have long fallen into the public domain), and the videos are of me playing the piano. Companies are allowed to do this because they will claim (via the automated matching) that it's too close to their own recording, and/or because they're a music publisher and it's too close to one of their editions. So every time an automated matching notice pops up I have to manually dispute it and respond that the video is of me personally playing, not a recording, Chopin's works are in the public domain, and I'm playing from a public domain edition. Every time the dispute goes through and they remove their claim; sometimes it goes through soon as I submit it (i.e. no delay for manual review) which leads me to believe that they purposely over-claim and see which suckers don't know the system and allow them to put ads up all over their videos and profit from it. Unfortunately I assume that's part of how Google justifies keeping Youtube running, so I doubt the system is going away.
Anyway videos from Sochi, unless they're fan cams, are obviously copyrighted so the networks do have the legal right to do it. There are some ways of defeating the automated matching (such as rotating the video) but the networks obviously see a reason to exert their rights -- which likely boils down to, they figure they can make money from it. That's what it comes down to in the end.