Hello, Chrilles.
I think i solve your problem. I wrote script as you need on Python using bs4.
There were some troubles with access to your site using 'requests'. It was forbidden.
So i decided to use 'Selenium Webdriver', which is much slower, but it's emulate real browser.
I checked about 300 internal links to test my script and found only 6 videos.
But it's already night, so i can do all job tomorrow if you want.
Please contact me and i'll show how it works or make any changes if you need.
I'll be in touch about 14:00 - 15:00 (UTC+2, Kharkiv, Ukraine)
Regards,
Pavlo
P.S. I think i know how to force 'requests' work. I will try it tomorrow.
P.S.S. Already did. 1768 internal pages. 37 youtube ids.