That's how every streamer works. Their core competency isn't building data centers, it's content. Everyone goes to the major cloud providers for this so they don't have to construct data centers all over the world for a geo-diverse content distribution network. Generally they'll have some sort of edge CDN of their own (colo'ed at major data centers in various regions, not that they've built out their own data centers for this, but with their own bare iron in racks) which will then push out to the big cloud providers and then individual streaming clients pull from there. I'm not positive on the details of the history here but I *believe* in the earliest days of Netflix they ran everything on their own servers with somewhat limited geo-diversity. Back then there were also news stories about people predicting that Netflix would "crash the internet" as bandwidth was getting saturated specifically with Netflix streaming. They started pushing more to a wider CDN across the globe that was still their hardware and then the stories shifted to net neutrality debates and FUD about ISPs charging customers add on fees for Netflix access while in the background trying to beat up Netflix for peering agreement surcharges given that traffic was heaving uni-directional. Obviously the sky never fell but Netflix also started moving pretty heavily back then to cloud CDN rather than their own boxes in colo's (I think they were big into Akamai back then as a CDN partner). In any case, their model of "in house" (although, again, colo'ed) edge CDN tiered out to major cloud providers has been the approach that every other streaming provider has since picked up.
tl;dr yeah, Disney can't support all of their customer's streaming needs from a network they personally own but that was never their intention and would be silly to attempt. Cloud providers do this, allowing massive scalability, and allows streamers to focus to their primary mission.
OK, back to funny pictures...
