It’s much simpler to use the server’s recording. Nobody has to press anything at all. It just works. (starting and stopping is based on absence/presence of silence)
Also, each performer’s recording is going to have different time codes. There’s an argument to be made that the server version is the canonical one. But that’s arbitrary, eh?
Much comes down to personal preference, but there’s nothing difficult about using the server version (aside from the tiny gotcha I mentioned that is easily overcome).
I’ve actually been tempted to play into a NINJAM server all of the time, just to get an automatic archive of everything I play without ever having to worry about hitting record.