Take a reading on the battery posts with a voltmeter and see what the voltage is with the engine not running. Then start the motor and take a second reading to see if the charging system is working, The reading with the motor running should be something more than it was without the engine running, let's say one volt more, a little over a volt is OK, and with smaller unregulated outboards 14 or more volts is not unusual.
If the reading does not go up or goes down, you have a charging issue, the failed tach is a first indicator of this because the tach runs off the AC output of the stator, which provides the AC to the rectifier, which changes it to DC to charge the battery.
My Johnson motors pick up the AC for the tach, before the rectifier or rectifier regulator comes into play, so a failed tach with no charging would suggest a stator problem on those motors, your Merc might work differently though.