You acknowledge that it's not that simple:
> running ffmpeg at a few quality settings upon uploading (there are maybe 3 articles with a video per day, so any old server can handle this)
Can any old server really handle that? And can it handle the resulting storage of not only the highest-quality copy but also all the other copies added on top? My $5 Linode ("any old server") does not have the storage space for that. You can switch your argument to "storage is cheap these days," but now you're telling people to upgrade their servers and not actually claiming it's a one-click process anymore.
I use Vimeo as a CDN and pay $240 per year for it ($20/month, 4x more than I spend on the Linode that hosts a dozen different websites). If Vimeo were to shut down tomorrow, I'd be pretty out of luck finding anyone offering pricing even close to that-- for example, ScaleEngine charges a minimum of $25 per month and doesn't even include storage and bandwidth in their account fee. Dailymotion Pro offers a similar service to Vimeo these days, but their $9/month plan wouldn't have enough storage for my catalog, and their next cheapest price is $84/month. If you actually go to build out your own solution with professional hosting, it's not gonna be a whole lot cheaper.
Obviously, large corporations can probably afford to do their own hosting-- and if push came to shove, many of them probably would, or would find one of those more expensive partner options. But again, you're no longer arguing "it's just an HTML tag." You're now arguing they should spend hundreds or thousands per year on something that may be incidental to their business.