The DGS has always reminded me of a bit of a back up generator solution. Great to know in your mind that if the power goes out you'll be back up and running again at the push of a button or will you?
Banking is a con trick - if everyone turned up at the bank looking for their money, most would be disappointed and the bank would collapse
The DGS scheme is a cover to try and ensure that does not happen but ultimately the state would have to decide whether to step in and throw taxpayers money in and hope to recover it by selling the bank's assets (loans) or by recovering the loans over time.
Bank runs are always possible but isn't that part of the risk of running such a business, rather than a con?
The DGS makes sense to me at least at trying to both prevent loss to depositors should the worst happen but also lessen the chances of it getting there by lowering the risk of a run.
I don't think there's any other motive from anyone's perspective.