I used this guy's script (https://github.com/ItsDeidara/DNSwitch) with an Apache server running on Ubuntu.
It does the exact same thing in TV mode. I haven't tried lower res. video's yet though.
If you haven't tried lower resolutions then most likely the partially implemented browser for router authentication doesn't have video playing capabilities (from a dev's perspective, why would it?)
at the very least this should be re-titled "The Switch Wifi settings browser crashes when loading a video"
at best, it shouldn't really be reported altogether
I wouldn't take this as concrete evidence that the Switch flat out can't do it. Even the Wii U could playback 1080p from the browser. Wait for the real browser to come.
I used this guy's script (https://github.com/ItsDeidara/DNSwitch) with an Apache server running on Ubuntu.
It does the exact same thing in TV mode. I haven't tried lower res. video's yet though.