For that exact reason I've had a weekly "emails and chill" date with a supporter. Brought coffee for both of us. I flicked through incoming support emails for 30-40 minutes....
It really depends. The per seat cost for ZenDesk is pretty low; if you have an efficient process, it can be a lot of effort for them. My work's support people are very efficient, so we were processing tons of tickets and bogging down their infra. We ended up building our own ticketing to get the performance we need.
On the topic of reading support email; I think it's important that developers regularly have tickets escalated to them, and regularly hear about the trending issues; but that doesn't mean they need to spend a day in the queue. Processing support email is a skill that needs training and practice to do well; it's not something an engineer can do once a month or a few times a year with good results. Of course, this may depend on the scale of your support queue and of your team; there's a big difference in a few emails per developer per day and thousands per day.
"My work's support people are very efficient, so we were processing tons of tickets and bogging down their infra"
That kind of says it all... There should be no reason support tickets should bog down their infrastructure. We're not talking about millions of real-time transactions here, even in the most popular consumer products. I should expect perfection because it's simply the most trivial product you can build, technology-wise.
Cost is one issue. Another is that these tools are designed mainly for triaging conversation volume and aren't really designed for PMs, salesfolks, or engineers to extract what they need from historical conversations.
Companies that want to get every team engaged with customer feedback can:
(1) Invest in their own data extraction and pipeline to push conversational info into cross-team dashboard.
(2) Perform or outsource a bunch of research and report on "voice-of-customer" to the rest of the org.
(3) Use active feedback reporting tools like ProductBoard or NomNom to organize (one kind of) customer feedback.
(4) Try passive conversation analysis from new tools like frame.ai (that's me!) or scopeAI
Then it is a product development priority to pick something else, just as you wouldn't pick a bug tracker, version control system, or language runtime that wasn't affordable to let every developer use.
Our product www.fwdeveryone.com is designed for this, if anyone is interested. We make it easy to share specific threads within private repositories for your business for these kinds of use cases.