I really don't like the model of throwing support over the wall to some team that didn't write the software. There is no incentive to improve the software under those circumstances, and the people that get paged aren't in the position to fix what's broken.
If you can fix the problem at 3am without thinking, then so can a computer program. Write that computer program.
If you can fix the problem at 3am without thinking, then so can a computer program. Write that computer program.