if you long-press a link to a video file to try and save it to your device (I occasionally do this while ripping videos from Twitter), you get the long-press context menu you want (with the "download linked file" item) for like half a second before the full screen video covers up your whole screen, and when you swipe it away the context menu is gone.
Is it possible that this is related to the iOS link preview feature?
When you long-press a link, do you get the full destination page preview? That might explain why the phone renders the page, including the video taking over.
Long-press on another page link (like one here on HN), and you have a tiny option at the top-right for "Hide preview," select that. Then try another link that has a take-over video, and see if it no longer does that.
If you hate video takeovers more than you like link previews, this might solve your problem.