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.
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.
That's true if you assume a constant rate of incidents. But when engineers are responsible for customer issues, they become aware of problems and can change the product to reduce those incidents.
In services like this one, letting engineers do the customer support is the closest they can get to eating their own dog food.
If incidents are handled by another department then it's more rewarding to implement another bonus-relevant goal than to reduce the number of problems.
Sometimes the support experience matters a lot as well.
I think, for what it's worth, that the IRC channel was maintained by Stripe employees on their own accord, not a company sanctioned support channel, and the help there was very "best effort".
Gone are the days where all the engineers were intimately familiar with every product and could offer you their tribal knowledge to resolve your silly issue.
Stripe is now a big corp. and there is no incentive for engineers to waste time on HN resolving customer feedback. There are proper channels that are better equipped to do so, namely the "clueless customer support agents" you speak of.
If your issue is truly some bug in the backend and not user error (which it usually is) it may eventually get surfaced to engineering.