MOV is a generic container. There's no way for CB Gallery to verify it's even a video or if it's encoded in a format that can playback in HTML5 video tag. Even then it typically requires Quicktime to even playback which anything other than Apple products typically does not have installed.

CB Gallery does not have a transcoder in it because 99% of our users servers will not allow for it as it's resource intensive so it can't transcode automatically for you. You'll need to convert the video (ideally to MP4 or M4V) on your device before uploading. Have added a feature ticket for CB Gallery to use ffmpeg to transcode if it's installed to help with this, but installing ffmpeg once this feature is implemented will need to be discussed with your host.

forge.joomlapolis.com/issues/7606

Nope, there are no plans to implement cloud processing like that. That's a significant amount of work that hardly anyone will use. The plan is to implement support for ffmpeg if your server has it available so your server can process the video. This will not work on budget hosting, but won't be difficult or time consuming to implement and is more open for users to use since it typically just requires upgrading your hosting package.

It's a major feature and it's already scheduled for the next major release. That's not something to be implemented in a minor release.