In theory this can work in a similar way to YouTube: list VODs and embed a player. There's a couple things to note though:
~There doesn't seem to be an efficient way to receive updates, only polling. Twitch supports both "EventSub" and "PubSub" but they seem to be focussed on stuff like subscription notifications and chat stuff. API access is a lot less strict than YouTube at least, there's no quota.~ EventSub does support a stream online webhook, which could be a good time to create the video, and just handle live/VODs like YouTUbe.
VODs include both "past broadcasts" (full streams) and "highlights" (labelled sections). Past broadcasts will be quite long, but is probably the more consistent way to go.
~The embed player does seem to have an option to include chat (which is the main reason to watch on Twitch instead of YouTube), and a JS player API similar to YouTube, so in theory should be fine but haven't actually tested it yet.~ Embed player specifically doesn't support chat replay, for some reason, which reduces their usefulness.
In theory this can work in a similar way to YouTube: list VODs and embed a player. There's a couple things to note though: