A live is not a file. X cuts it into thousands of small pieces and sends them as they are made, so there is no finished video sitting anywhere — not on X's side, and certainly not on ours. While the broadcast runs, there is nothing to hand you.
Once it is over
When the broadcast ends, X often republishes it as a normal video post. From that moment the link behaves like any other: paste it, pick a quality, save it. This is the case worth waiting for, and it usually appears within minutes.
If the author did not keep the replay, nothing can bring it back. The same applies if it is still sitting at a /i/broadcasts/ address rather than a /status/ one.
What stitching a replay costs, honestly
Some sites do offer the replay anyway, by downloading every piece onto their own server, gluing them together, and sending you the result. It works. It also means the whole broadcast travels twice — once to them, once to you — and an hour of video can be one to three gigabytes per download. We measured one: the first bytes arrived after nine to thirteen seconds, with no size announced, no progress bar, and nothing to resume if the connection drops.
We would rather tell you what is true today than spend that on every visitor. If we add replays later, it will be with the length limit written on the page and the size shown before you start.
Spaces are audio
A Space is a live conversation with no video at all. There is nothing for a video tool to pick up, and when a host shares a recording it arrives as its own kind of post. If what you want is the sound of a normal video, that we do have: choose Sound only in the list of formats.