Is it allowed to embed YouTube videos behind a paywall?
You may not use embedded YouTube videos behind a paywall. That is prohibited by YouTube in section 4 of the YouTube terms of service:
D. You agree not to use the Service for any of the following commercial uses unless you obtain YouTube's prior written approval:
- the sale of access to the Service;
Youtube does make exceptions for using the embedded player on advertising supported sites, but there is no exception for paywalls.
You may not use a third party player for YouTube videos. This is also from section 4 of their terms of service:
A. You agree not to distribute in any medium any part of the Service or the Content without YouTube's prior written authorization, unless YouTube makes available the means for such distribution through functionality offered by the Service (such as the Embeddable Player).
B. You agree not to alter or modify any part of the Service.
C. You agree not to access Content through any technology or means other than the video playback pages of the Service itself, the Embeddable Player, or other explicitly authorized means YouTube may designate.
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F. If you use the Embeddable Player on your website, you may not modify, build upon, or block any portion or functionality of the Embeddable Player, including but not limited to links back to the YouTube website.
Streaming the video through another player would violate the prohibitions on making the videos available other than the ways provided by YouTube or the prohibitions on modifying their embedded player.
I'd suggest looking at using another video service such as Vimeo. They have relatively inexpensive packages that can be used with commercial service.