> The concept of property represents a set of rights (and maybe obligations) that an owner of said property does possess.

Just the one right, actually: The right to decide how the property is used. That's both necessary and sufficient. There are no obligations beyond reciprocation, respecting that same right when it comes to others' property.

> Your nose is not your property by the way, because the rights you have towards your own body are very different (and a lot stronger) than the ones you have towards your property.

No, they're exactly the same. Your body is your property. The only difference is that you can't give up ownership of your body as a whole (it's inalienable), and that's simply because there is no practical way for you to relinquish control over your body to anyone else short of your death. Even if you agreed to it you would still remain in control, which would render any such agreement void. Certain individual parts, of course, are a different matter. In every other respect your body is just like any other kind of property.

> What they actually might do is be in conflict with the rights of the owner.

Any use of others' property against their wishes conflicts with the right(s) of the owner. This is a distinction without a difference.

>> There are no significant barriers to entry in starting a video streaming site.

> Of course there are and you even state one of them three sentences later.

DMCA complaints are a problem created by the government. If you want to do away with copyright, I have no objections. However, this is not a significant barrier to entry because it only starts to become a burden at scale. Small sites with user-submitted content regularly deal with such matters manually, while large ones have had time to implement the needed infrastructure for automating the process.

> ... starting such a site will not change anything about youtube's monopoly.

Because YouTube doesn't have a monopoly. What it has is popularity. Anyone can start up a site that does what YouTube does, but that won't automatically make it popular. The responsibility for that is 100% on the prospective competitor.

Popularity is fickle. Sites everyone turned to yesterday may be deserted wastelands tomorrow. (E.g.: MySpace) If your site is actually better at giving people what they want, it will win. Your problem is that YouTube actually is giving most people what they want. If you broke it up you'd end up with a bunch of YouTube-clones with basically the same policies, because they're trying to attract the same audience YouTube has now. You want to change what people want, which is naturally unpopular and likely doomed to fail.

In short, despite your complaints about "natural monopolies" and the difficulties of competing, you don't really want to compete with YouTube, just leverage its existing popularity to push your own agenda. The problem with this is that its popularity is a product of its audience-pleasing policies, not any sort of monopoly, natural or otherwise. If you did manage to force YouTube to follow your preferred policies it would become less and less popular until it was eventually be replaced by a competitor that looks more like the YouTube of today. So breaking it up wouldn't help you at all. Regulating it and actively preventing competition, or regulating all such sites the same way regardless of "monopoly" status, would better serve your purpose, but that's just exchanging one (non-)monopoly for a much larger monopoly, namely the government.