You need somebody actually in charge of prioritization; often this is not the engineer, but the PM.
But like all humans, different project managers have different strengths and weaknesses. Some PMs are really good at certain things like schmoozing the stakesholders and keeping them happy, but aren't the sharpest knives in the drawer when it comes to actually building products, understanding technical details, and solving business problems.
Good ideas can come from all corners, but I'd say it would be a good thing for the engineers in particular to experience some customer service tasks for some of these reasons.
1) They have better insight into how hard something would be to fix and can probably prioritize a little more effectively in some cases.
2) They might not be aware of the problems that a particular issue causes. In many companies, even those with good communication between departments, it can be easy for something important to not be communicated well enough even with a great PM.
3) Getting exposure to some of the pain of doing customer service might give the engineer more empathy about the problems that customers and staff have to deal with and might have them take it more into account when building future features for the product.
Paying a good engineer for a day to deal with customer service issues? ~$1,000.
Having the engineer directly learn about the pain points that exist in the organization and giving them real incentive to finally fix the problems that cause the entire company endless grief? Priceless.
I don't know much about Stripe's organization specifically, but in general maybe it's part of the problem that engineers don't have an occasional rotation between projects assisting with some customer support tasks and learning about the real pain points and the damage that they can cause to your business's reputation among customers. It's very easy in technical organizations (I'm looking at you Google) for the engineers to just magically ignore the numerous flaws and holes in their processes.