There are definitely advantages to capturing in 4K, even if you're only going to output to HD. For a start, by using a 4K image as your source, your final HD image will effectively be four times oversampled (two times in terms of linear resolution) so your colour resolution will be better and there will generally be more information in the picture. This really is true - we've seen it many times!
And then there's the fact that you can zoom in and then pan around the image. It's a major feature for sports broadcasters (and one reason, perhaps, why they might adopt 8K before the rest of us) and it can actually lower the cost of production because you don't need as many cameras.
Here's another version of Andrew Dean's music video which he shot originally to compare several different cameras. Here's the same performance, seemingly shot from several different devices, but actually just one: a 4K Panasonic GH4.
Note that the camera didn't move at all in this production. All the moves were programmed afterwards. Sometimes it's hard to believe because the moves are so smooth and natural!
Note that in this case the file was output as SD, which has freed up even more movement in the "virtual" camera position. The quality's not as good but for output to the web, this seems a perfectly valid thing to do.