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Yes, we do this; it seems to be both the correct solution, and one which works effectively in practice -- use email properly.

Auto responders generally send their messages to the mail Sender. That seems correct to me. We send automated mail from robot@, with a Reply-to of support@ -- because that's logically what's happening.

All of bounces just end up back at the robot, and IMO that's where they belong (and potentially can even be dealt with automatically). The vast majority of out-of-office do that too. And humans end up emailing support.

The name 'robot' was chosen to gently imply to the user a few things.