Can you download a live stream?
Not while it is live. The reason is simple: there is no finished file yet, only a broadcast still being produced.
Why live is different
An ordinary video is a complete file sitting on a server. A live stream is a sequence of small segments produced as the broadcast happens, with no defined end until the broadcaster stops.
A download means fetching something complete. During a broadcast there is nothing complete to fetch, which is why a download request on a live video is refused rather than queued.
After the broadcast ends
When a stream finishes, YouTube usually publishes it as a normal video at the same address, and from that point it behaves like any other upload.
Processing a long broadcast can take a while, so a link that still reports as live shortly after the end will generally work once processing completes.
When the replay never appears
Broadcasters can choose not to keep the replay, and some streams are removed as soon as they end. If no recording is published, there is nothing to download afterwards.
Many channels post highlights or a re-uploaded version separately, which is usually the practical answer.