While having demos here would be nice, I believe it would require a tradeoff with negative consequences for the actual game releases. GOG is a small company with limited resources. They don't have a lot of employees, and quite frankly customer service and tech support often seem overwhelmed as it is. It's not because they are being lazy, though, they just don't have enough manpower. Diverting their attention away from the games themselves would make the situation worse.
Offering demos here would require substantial effort. The demos would most likely not be provided by the publishers, so GOG would have to track them down, make sure they work, are not modified and don't contain viruses. They would have to negotiate the rights to offer the demos on this website, create an installer for them and provide support to people who are not even paying for it.
I would rather have the GOG team focus on releasing new games, ensuring compatibility and fixing technical issues. To me, demos are like a nice extra, but by no means essential. The games sold here are old, and oftentimes people have already played them before. Really, the worst that can happen is that you spend 10 bucks on a game you don't like. People have pointed out important ways how you can form an opinion even without a demo.
GOG can't really post links to YouTube videos on the games' store pages since the videos are user created and not in any way official. Can you imagine Steam linking to a user-created video? You can often find interesting links on the forums, though. The problem with linking to demos for old games is that they are almost never hosted on the publisher's servers anymore. They are often uploaded by fans, leading to many of the problems you have listed yourself. Technically such demos might be piracy, might contain malware etc. Linking to them would require verification in each case that the downloads are fully legal and free of problems, which is probably not feasible.