Youtube has (or at least used to have) contracts with ISPs where they'll store copies of the most popular videos locally to reduce load. Obviously you can't do this with live video, so arguably Twitch is in an even worse spot than Youtube.
This is complicated.
The dynamic is that the way to improve service and reduce costs for most people involved used to be direct peering. You throw me a fiber in a data center for 'free' and I'll serve the traffic. This still happens quite frequently, but it breaks down in the face of 1: The colossal demand for video on today's net, 2: The attempts of Verizon, Comcast et al to turn that model on its head and charge for peering to get good service (what caused the original Net Neutrality debate) and 3: The desire to 'push down' the video serving as far as possible.
Netflix was sort of first out of the gate with the 'local cache' idea. They'll give you servers for free to host on your network to do what you're talking about. This works quite well but for a couple years now the ISPs are getting sick of that business model - they're on the hook for the power, cooling and care of those servers, which provides a benefit to Google/Netflix/Amazon. Some of the big boys (Comcast for example) don't play the game at all for this reason. Others are going for more radical approaches. I've been around projects to put caching servers into neighborhood-level CMTS cabinets. Recently they came around my neighborhood and put a bigger box for xfinity up on the pole - I suspect this is to enable an effort like that.
Local caches don't HAVE to break down where live streaming's involved. Once upon a time multicasting was supposed to fix this problem on the Internet but that sucked at even moderate scale. Today, you just tell the browser's player to go to a local cache that rebroadcasts the stream for people in your area. It's not that hard. It
is harder for Twitch because other than big-name streamers the effort isn't worth it.
Notably Amazon about a year ago did a huge, high pressure campaign to get everyone to massively beef up their peering with AWS. This was to support their Prime Video exclusive Thursday Night Football games. This was not - so far as I saw in traffic analysis - rebroadcasted in local caching but I suspect this was the strategy. Cloudfront, AWS' CDN, has always been kind of an also-ran, and they wanted to beef that up. That effort is a big boost to Twitch who IIRC
still runs a hybrid AWS/local data center model ten years after acquisition.