Currently, Studio One 4.6.5 treats the Video Player as a completely separate thing from the mixing process, so much that if you're on Windows using Windows Audio, muting the video will completely mute Studio One (all of its tracks), which defeats the whole purpose of the feature.
Even if one does extract the audio from the video player into an audio track, you will still have two sources of sounds - the video player and the extracted audio track - which are not synced to each other, and muting one will not mute the other.
Video's audio tracks should be shown and treated as mixer channels after importing the video so we can control the volume, panning, muting and soloing, and add VST/native effects to them even if we can't edit them (to keep video sync).
Presonus' support says that the fix for this issue is to install third-party driver (nightmare) ASIO4ALL. There's need to be a native solution to this issue, not ASIO4ALL.