There seems to be a lot of smoke and mirrors in the diagnosis. Very few shops have the anode/cathode cell tester or a qualified mechanic to properly use one to definitively diagnose a dead cell. The grab it and test it at the front door testers often say DEAD CELL when the battery cannot produce rated amps. Whether at Meineke or any other shop if the alternator is bad and the battery has been severely discharged repeatedly both would need replacing to have any certainty that the vehicle would crank the next morning. But the local Meineke has a poor reputation. Can it be a coincidence?
Unlikely your alternator is bad. Usually the red “battery” comes on when your alternator isn’t generating enough current to recharge the battery, and it sounds like your “battery” light is NOT illuminated until the battery goes totally dead.
Just get a new battery. Most likely that will fix the problem.