Cron job log reads 'No MTA installed', does that prevent the CRON job from finishing?

No, it does not prevent the CRON job from finishing.

Normally, you shouldn't let your CRON job generate output without telling it where to write it. If you do, like in this case, CRON tries to mail it somewhere (also specified in the crontab file). If it doesn't succeed, it discards the output, as it says in the log file, and goes on.

In your case, you don't have a "Message Transfer Agent" installed. This is a generic name for packages, like postfix and ssmtp, that provide a mail command.

I recommend redirecting SYSOUT and SYSERR from the rsync command using > and 2>, respectively. If you're not interested at all, redirect output to /dev/null.