I agree with it's a starter issue. Check the connections to the starter & starter solenoid 1st, make sure they're clean & tight. If they are, likely the starter is on it's way out.
These connections are the only ones that affect the starter and not the rest of the electronics.
-If it was a bad/failing battery, everything else would die when you tryto start it.
-As mentioned, the alternator is just there to charge the battery.
PS, the parts circled are the starter solenoid & Bendix mechanism. The Bendix bit is what drives the starter gear out to the flywheel/flex plate to spin the motor over when the starter motor spins. I'm going to guess the parts guy meant "Bendix" and you heard "Bandits".