The problem with that is that you are having engineers who cost the company $75+ an hour spending time doing the job that can be done by someone costing ~$30 an hour. Also, you run the risk of annoying your engineers.
I have been part of too many engineering teams working on the wrong things because of a lack of customer interface.
I've been in the situation where each developer would take a day of the week handling support. But the support requests could be for any of the software built by the entire development team, not just the things that the individual developer worked on.
So then in many cases the developer was acting as an extremely expensive email/call filter. And the only lesson being learned was that some of the other developers weren't very good at their jobs.
Also, consider that engineers can fix problems faster.