Rotating shifts on customer service having them serve as a customer service rep for a day 1 to 4 times a year.

Yes, as long as they also have visibility into everything else customers do where things go smoothly. You definitely need to see and fix problem areas of the product, but if 95% of the product is working well, you can get a false impression that the sketchy 5% is a much larger piece of the puzzle than it really is.

Good idea. We should also let customer service write code from time to time, so they get familiarized with how the product is built and get a sense of what's feasible and what's not.

Yes ! nothing like doing support for basic questions to understand the pain points and what can be improved.