Definitely a few factual errors here that ought to be corrected.
On day one of the acquisition, Youtube's egress network was at least 4x the size of Google's, built and run by two guys. This shouldn't be a shock, you need a lot more bits to serve video than search results. For the hottest bits of content, third-party CDNs were serving videos and thumbnails.
There was no collapse imminent, but there were concerns about getting YouTube and Google infrastructure on a common path. BandAid was named as such because the goal was "not to break the internet." It was a small project with maybe a dozen members early on, all solid people.
YouTube had its own contemporaneous project, née VCR - "video cache rack". We did not necessarily believe that BandAid would arrive in a reasonable amount of time. Generally Google has two versions of every system - the deprecated one and the one that doesn't work yet.
VCR was a true YouTube project, 3 or 4 people working with one purpose. It was conceived, written, physically built and deployed in about 3 weeks with its own hardware, network setup and custom software. I believe it was lighttpd with a custom 404 handler that would fetch a video when missing. That first rack maxed out at 80Gbps a few hours into its test.
After several months, Bandaid debuted. It launched at ~800Mbps and grew steadily from then on into what is certainly one of the largest networks in the world.
YouTube mostly moved things to Google based on a what made good engineering sense. Yes, a few of them were based on what would break next - thumbnails leaps to mind. Search, which we thought was a no-brainer and would be "easy" took more than a year to migrate - mostly due to quality issues. Many migrations were purely whimsical or some kind of nebulous "promo projects." Many more stayed separate for more than a decade. When a company gets large enough, the foolish consistency tends to trump a lot of other engineering arguments.
To the ancestral poster, do not despair. You can transcode, store and serve video, as you've likely surmised it's not all that difficult technically. In fact, it's so much easier now than in 2005.
What makes a great product is hard to describe and not always obvious. The economics will depend on your premise and success. "the cloud" gets pricey as you scale. There will be a long path of cost optimization if you get big enough, but that's the price of doing business at scale.