I checked, video is perfectly played on the desktop (with bg image). It all depends on the capabilities of your Internet connection and the size of the video file. By the way, the video file should be optimized as much as possible before uploading to the site. My test file is 10 MB 34 seconds.
Played in Mobile Chrome, in Mobile Firefox you need to click the Play button - all this in 4G networks. In 3G and less, not played.
In general, different mobile browsers have policies of varying degrees of rigidity over video playback. So it makes no sense to build a mobile design using the background video. Also, the background video consumes too many resources for the mobile Internet, which leads to slow speed of the site and as a result, the loss of potential customers. For mobile devices, the background video should be off at all.
And I know how the poster attribute (it is not tag) works. Just like the background image. There is no javascript to keep track of video download. The image is simply under the video and when the video booted, it starts to overlap the image.