Videos uploaded via the API are not showing up on video.twitter.com. Here is an example video tweet: https://twitter.com/omgimfestive/status/725331011807133697. This was uploaded via the API and the tweet object as seen through the API shows up as a video. It displays correctly and shows up in the video analytics on analytics.twitter.com.
I’ll let someone on the Media Platform team confirm, but my understanding is that uploading video to video.twitter.com (if granted access) and uploading video through the API are two completely different products. video.twitter.com only has a record of videos uploaded through its in-browser interface and anything uploaded via the API is not intended to show up there.