The only way I have found is the Cueler App which has a relevant setting:
However, new uploaded videos are still not shared on the fan page and I cannot reproduce why.
Are there any alternatives that I could try?
YouTube – How to Autoshare Uploaded Videos to Facebook Fan Page
facebooksharingyoutube
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Best Answer
This IFTTT recipe Post YouTube upload to Facebook Page should do the trick. You'll need to create an IFTTT (stands for If This Then That) account if you don't have one and add your FB Page and YouTube as "channels" but that's not nearly as difficult as it might sound and you'll be able to customize formatting etc. for posting your vids to the page, among other things.