Q: 07 Cobalt ignition was replaced on a recall. Now it won't start.

07 Cobalt ignition was replaced on a recall. I took it less than a quarter mile. I parked it and left it for a few hours then took it home also less than a mile away. Went out this morning it wouldn't start. Called every mech/tech I know, was told to take it back to the dealership because someone fucked it up. Called the dealership told them what was going on. It doesn't click or kick, I have power it just won't turn over. Their mech/tech told me my batt didn't have the juice needed or it was the aftermarket remote starter(in when bought never used by me), that they know what they're doing. After waiting about 20 mins I tried it again, it turned right over. I drove it around for maybe 15 mins. Parked. Went to get my kid off the bus, it started. Tried to drive home, wouldn't start for 15mins. Mech/tech brother tested batt, as well as hooked up the jumpers and revved up just in case it tested wrong. It registers a little over 12v before we hooked the jumpers. What happened to my car?

My car has 130400 miles.
My car has an automatic transmission.