On the current theme, there was no page.php file present, so WordPress was reading index.php and, according to the theme’s instructions, displaying the_excerpt(). It will strip many tags, including <video>, and that’s exactly what was happening.
The content of the posts is is displayed based (mainly) on the contents of the file single.php, but for pages it is (mainly) the file page.pgp. (For more detailed information, everything is well explained and documented in the Template Hierarchy section on the Codex).
