The trade-off you make when using short IDs is that you can't generate them at random. With 128-bit Id, you can't realistically have collisions, but with 64-bit ones, because of the birthday paradox, as soon as you have more than 2^32 elements, you're really likely to have collisions.
However theft of the encryption key is a concern, since you can't rotate it and it just sat there in the code. Nowadays they do something a bit smarter to ensure ex- employees can't enumerate all unlisted videos.
Random 64-bit primary keys in mysql for newer videos. These may sometimes collide but then I suppose you could have the database reject insert and retry with a different id.
Surely if it was a great chore for YouTube to have random-looking int64 ids, they would switch to int128. But they haven't.
I'm a big fan of the "works 99.99999999% of the time" mentality, but if anything happens to your PRNGs, you risk countless collisions to slip up by you in production before you realize what happened. It's good to design your identity system in a way that'd catch that, regardless of how unlikely it seems in the abstract.
The concept of hierarchical ids is undervalued. You can have a machine give "namespaces" to others, and they can generate locally and check for collisions locally in a very basic way.
UUID generation basically has to use a CSPRNG to avoid collisions (or at least a very large-state insecure PRNG).
Because of the low volume simply using /dev/urandom on each node makes the most sense. If /dev/urandom is broken so is your TLS stack and a host of other security-critical things; at that point worrying about video ID collisions seems silly.