Why can't we view other people's highlights?

Highlights are saved to the user's computer, so with the current method it'd mean hosting it on the server instead and taking up a lot of space that costs a lot for Riot. They aren't like replays which take up very little space, they are video data which is very large, but it can be watched at any time with any number of video players without needing the client.

Additionally, a replay is just spectator data, meaning it uses the client's information to figure out what's going on in it, and it needs the same version of the client it was recorded from to know how to play correctly (if X item no longer exists in the current patch, and older replay with that item wouldn't know what it is, for example, or an updated ability would behave differently in the replay than it did in the actual match and it'd break).

This is probably something where a compromise can be made and I think it's worth exploring, but it's a lot more work on Riot's end than simply "show the highlight in their profile".

The easiest way to go about it would be to reformat it onto a site that supports the media file (I'm lazy and use Facebook), and link it from there. It's what I've been doing since the release of the feature, and so far I've yet to feel too hassled by it.