That's a trick (and a completely different) question from:

If I want to distribute billions of videos to billions of people on my own site, that’d be great, but my options are basically to pay Google, Amazon, or Microsoft for help.

If you assuming this is a load spike well above your maximum throughput, you just fall over. That's the reason the cloud took off in the first place; if you do it the old fashioned way, you can get overwhelmed and have no recourse but to fall over because you can't provision servers fast enough.

If this is a "normal" occurrence, you're probably talking about round-robin'd DNS distributing to (hopefully) the nearest data center where you have a load balancer, then that could hit another layer of load balancing on the machine or directly go to servers on the machine depending on how you have things setup, then that could hit several different designs of web server with their own quirks (asyncio -- non-preemptive multitasking --, threaded -- preemptive multitasking --, single threaded, or some mix thereof depending on your design -- pros and cons to all of them).

Those could do several things depending on how fast your data centers become consistent WRT uploaded videos. We'll just assume you already have a copy of the video in each regional data center. You fetch from there and serve the file, like any other file.

... or maybe you get really fancy and use WebRTC to get the files from clients already watching the video.

But please, add another surprise requirement and continue to strawman.

EDIT: Oh neat, found out after the fact PeerTube actually does that WebRTC trick https://docs.joinpeertube.org/contribute/architecture#the-peertube-player :)