Running Youtube on a peer-to-peer basis would be a nightmare. Either whoever had the video uploaded would have to have their PC on at all times (as well as having a very hefty internet connection), or else you'd have to hope the video you're trying to watch had somebody available to upload each segment in order at all times. Guy who has the 53:21-53:26 segment of the cooking video you're trying to watch is offline? Guess you'll never find out what the brownies look like.
A centralized server is the only way streaming video can work.
Something worth noting in this hypothetical is that streaming torrented videos while downloading is already something that people frequently do (on protocols that were not designed with streaming in mind) and one would see in the client whether the video is being seeded (and would make a decision to steam or download accordingly if the video is obscure enough to have 0 current seeds). It is true that the experience may not be as consistent as what youtube offers for every single video, but there's little reason to believe the experience for videos that aren't both obscure and from a defunct source wouldn't be comparable to what youtube offers.
Gradually pay for that, and it quickly becomes obvious that the most effecient way to serve the content is to have it all hosted on one giant server farm. You've just invented the current Google system from first principles.
In the peer-to-peer system, most people who would be serious about making videos (like, making a living out of it through embedded advertisements or donations) would probably end up paying $5-15 a month for a seedbox (at current prices) to maintain constant availability, which would indeed have a chilling effect on new uploaders (who yes, could also keep a cheap machine on to seed off their home connection as a "budget" option). It seems to me like a difficult question whether this would actually be more or less efficient in the aggregate in terms of electricity, hardware, and labor than what a company like youtube ends up doing to maintain the centralized model. Bear in mind that the hosting process could just as easily benefit from similar economies of scale that youtube benefits from, and that it isn't just one huge server farm that they use, it's an enormous network of huge server farms that constantly copy data around to keep speeds as close to instant as possible (which is something the P2P system does naturally, at least if the video is popular enough to have someone else in your city seeding it).
At any rate it was inaccurate of me to say that all their operating expenses are a complete waste, since some of what they do would be pushed onto seedboxes.
But there's a bigger issue. I can easily find and start watching a video from 2006 on YT and start watching it. How likely do you think it is that people are going to reliably seed torrents for 15+ years?
I agree; it's somewhat unusual that we can have a centralized host that is so far willing to commit to hosting everything forever at no cost to the uploaders, and in a decentralized system you wouldn't get that maximal of a guarantee without nonprofits such archive.org committing enormous resources to it. Although I will add a non-critical point here that barring the apocalypse that particular example of an old video probably would be seeded forever.
One last thing to note is that P2P is also frequently used for streaming live broadcasts, if the idea of streaming P2P seems alien; as I understand it, in many parts of the world live pirating of cable TV is popular. There have also been efforts to make P2P social networks (i.e. alternatives to the facebook model currently occupying a natural monopoly). The point is that dependence on centralized servers isn't the only way of doing it, including having full support for user profile space, channels, searching, recommendations, etc.