The trick is that each link you extract from YouTube will only work for your specific IP address. You cannot hand-off this link to another IP address or it won't work. This is why "YouTube Video Downloaders" are all scripted client side - so that a visiting user accesses YouTube from their machine and gets a YouTube link tailored for their IP address.
I'm not sure a web-based proxy would accomplish what you want because the visitor to your site would have a different IP address than your server which would be accessing YouTube. You would have to find some way to tunnel their request directly to YouTube.
You would most likely have to download/stream the video to your server and then offer it to your clients.