Maybe it's just me, but as a PM I want to hear as much as possible from customers. Asking other teams to prioritize e.g. two requests ensured that I'd be missing potentially valuable customer feedback which made me twitchy.
It's not support or sales' job to decide what's important. It IS their job to tell Product what they're hearing a lot of (ideally with minimal disruption of their workflow). But it's Product's job to decide what's important. Not having that information leaves Product (and by extension the business) more in the dark than it should be.
Solving this problem is why I built Savio[1] to help support and sales teams quickly send customer feature requests from Intercom, Help Scout, and other tools to their Product team... and for Product to have a sane list with tools to prioritize their features (e.g. "Show me feature sorted by # requests, or cumulative MRR, or from Churned customers").