I'm kinda proud of this, but it took me way too long to come up with the right solution 😅 I also couldn't find anyone on the internet who wrote something like this, so I'm sharing it in case others find it useful.
This can be used to wrap an $output in any CLI command, it will intercept all the newlines being printed out and insert the current time at the start of every line.
Easier said than done, because if we just naively insert dates after each newline, then the timestamp of a line would be the time from the previous message, not the current line being printed. So to correct that, we need to keep track of whether the previous write inserted a final newline, and if so prepend a date; and we need to skip adding a date on the last newline of that write.