[bangster]

That setting (Auto Engine Off) is not related to the auto start/stop functionality.

That Auto Engine Off is not a permanent setting, it is only temporary. If you uncheck it (or disable/change the amount of time for it in the Vehicle settings in SYNC 3 screen) then next time you shut off the car and restart it, it will default to on again.

For vehicles that have intelligent access and push button start, if it idles for 30 minutes it will shut off the engine and also the ignition so the battery won't go dead.

If auto start/stop is always disabled (the grey A with a circle), then yes, resetting the BMS if you have a new battery can re-enable it. I was able to do it myself following the steps of resetting it:
don't put your foot on the brake
press start button
flash to pass 5 times (turn signal stalk towards you 5 times)
press brake pedal 3 times

Sometimes the battery light flashes to let you know it is complete, but not always

 

[bangster]

Also the process from your owners manual to reset the BMS is that it needs to sit for 8 hours, locked, with the ignition off. This didn't work for me, more than a week after replacing the battery my auto start/stop was still disabled (I think yours was because of the weak battery...), and even though I didn't get the flashing battery light after doing the steps above, immediately that time I drove the vehicle auto start/stop was enabled again.