Three reasons:
1. Picking something to watch takes time. Sometimes I only want to see something in the 15 minutes that I'm dining alone. My meal gets cold before I start the video
2. Choosing something to watch is stressful. If I'm tired and I don't know what I want to see makes me more tired and frustrated. These are the times that I don't want the freedom to watch I want because they are the times that I don't want to think about what I want
3. The random factor of watching something that I would never watch by myself it's something that makes me go outside my bubble. I can't say how many good movies (or songs, etc) I found by that randomness
I'm not against the freedom of streaming services but there are moments that I just don't want that freedom. So, thank you!
You can certainly fake it as a workaround. For example, you'll notice that "I'm feeling lucky" on Google simply follows the first search result. Streaming services could take what is already computed as the first result on the "Home" page and use that, for example.
But at that point why not just click on the first video? Unlike Google, which doesn't give you much until you enter a search query, all of the streaming services I know of have already given you your "lucky" matches by the time a "I'm feeling lucky button" could be presented. Two buttons side-by-side that do the exact same thing doesn't offer much.
The only response I have is that purposefully-clicking the 'random' button has a psychological effect over clicking the first video returned which (gut check) makes me think it will be more easily tolerated if it ends up being "off-beat" since I didn't explicitly click the first selection (thus choosing it).