Implementing this is a pain in the ass, any new intermediary (haproxy, nginx, ...) introduces a potential new source of buffering or IO loop that attempts to helpfully batch up your writes. But when the technique is implemented, it is legitimately an amazing way to reduce latency at the client.
I use this technique in an app where an opening chunk is used to ensure the browser has begun fetching/executing the app JS bundle before a list response is rendered. After the list is rendered (incrementally as each chunk becomes available) it is again used to execute a slow summary query without having to break out a separate request. All that fits in a single HTTP response body, no additional latency is introduced by making separate API requests for the list or for the counts. Because of the initial chunk causing the JS to execute, it is also possible for the app to display certain dialogs (depending on query string parameters) that are immediately ready for user input even before the subsequent responses have finished rendering or downloading. In the case of this specific app, the technique means some dialogs are usable 300ms earlier than otherwise
Another issue not touched on in the post is how all this newly unbuffered incrementally-arriving data interacts with visual rendering on the client, there is potential for a lot of flicker and burned CPU continually re-rendering UI elements.
It sounds like Airbnb are disabling compression to make their implementation work. So long as CPU isn't a problem, you can also implement compression in the backend, and flush the compressor each time some useful unit of work is produced (some task completes, or some amount of data e.g. aligned to the size of an Ethernet MTU is available for writing). Assuming all the stars align, this should produce compressed output in a form the browser can act on every time it manages to read any amount of data from the backend.
All this effort seems like nonsense until finding you have a perfectly functional app on a crappy airport wifi network where it was otherwise impossible to even get Google to load