This sounds great in theory, but in practice it turns into an asshole filter (https://siderea.livejournal.com/1230660.html). In other words, what you've just done is create a set of incentives that reward the most obnoxious people with your attention, and starve those who politely wait. In time, those who politely wait will write you off as a flake, and then you'll wonder why everyone who contacts you obnoxiously sends three e-mails, four slack pings and an after hours text message for good measure.
The issue with this approach is that there will be a bias towards people who quickly/persistently follow up. Personally, I'm very hesitant to do so especially with busy people, as I don't want to add to their workload (or indeed, I dislike being pushy in general).
As a PM, you need to listen to your entire community, not one person. Whether they politely request something and wait, or whether they are excessively persistent, that is still just one request. And unless you only have a handful of customers, you need some consensus from the entire community that the request is a good one before saying "Yes", and getting to work on it.
Please come host a seminar with my product management group. Maybe if an outsider/third-party says it they'll actually listen.
The most effective PMs I've worked under had a heavy filter on incoming requests, at least for established products, with "No" being the default answer. If multiple people started asking for the same thing multiple times, clearly it was important. But if you say yes to every little things that comes in via any channel, you end up with a complex, cluttered product.
I feel that a product like this is better applied to a customer services/support team, where you really do want to make sure every little complaint is addressed properly.