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Is it possible to do this using a youtube private channel and have the website login to youtube so that access can be restricted to members of the website ?
You'll notice that the first thing it does for you is log you in. Note that currently there is no public release of this module, but the code is in the CVS repository [cvs.drupal.org].
That might at least point you in the right direction for integrating Youtube with whatever CMS you're using.
I'm sure other CMS have released or will be releasing similar modules to take advantage of the public Youtube API.
btw - in .NET you can create this proxy class code automatically in vb or c# by pointing svcutil to the wsdl file
svcutil [youtube.com...] /out:myproxycode.vb /l:vb
More generally, going beyond drupal back to the root source...
The PHP client library can be used to work with either public or private feeds.
but you may find the non-PHP-specific docs more instructive:
[code.google.com...]
You'll probably also be interested in the documentation on custom feeds and private user favorites feeds [code.google.com]
See also the page on getting the Google data API PHP framework running (I haven't done this personally)
[code.google.com...]
In any case, I think the goal is to give full access to the API from drupal and that API definitely provides for client login and private feeds, but I don't know about private videos.