FOSS is fundamentally about freedom--the freedom to control the software you use.
Even if LinkedIn or Twitter were open source, I would still not be able to control them--I don't have a spare server lying around and I certainly don't have a way to get everybody else on my own system! So if I took LinkedIn's source and improved it, it would be useless: I would first have to get a server to run it and then I would have to get everybody I know to switch to it. This means that, even if LinkedIn was open source, I would still have to use their implementation on their servers; it would not have much of a practical effect.
Put another way, Reddit and HN are open source. But this does not mean I can meaningfully modify or control the code; this control is naturally left to the people with the servers. There is very little practical difference for me as a user between an open source Reddit and a closed source Reddit.
Now, this is not to say that web apps should not be open sourced--quite the contrary! Open sourcing is great for reasons besides freedom: you contribute to the common good, give more people insight into the development process and help other people with similar projects. But it is less critical than having open source programs that run on my own computer.
For one, I don't view services like Quora as being in the same class as "software I run on my own machine" in the first place. It's a subtle distinction, granted. But I've just never viewed those services that way.
Secondly, because there aren't adequate F/OSS replacements available. I can use Fedora Linux with KDE and get an experience that's better than Windows. I don't know of a Quora like service that makes their code available, that gives a better experience than Quora. That said, if such a thing were available, I would prefer it.
I don't see it as a subtle distinction. Simple using shouldn't grant rights. If I touch an ATM, or if I borrow someone's computer there's no reason I should be able to demand the source.
Free Software is about user rights. If you are fine with having your rights restricted by the software provider, then this kind of freedom is not for you.
But I like my own freedoms and I'll avoid buying software that prevents me from exercising them.
