Oof, I really don't think that's a good use of a product manager's time. I want them out talking to customers, or evaluating similar products from other companies, or using their own product to understand its capabilities and gaps, or looking at metrics, or just writing and debating product proposals. I honestly think there is already way too much work that is way too valuable for them to be doing, for them to take on project management duties. It's a much closer fit for an eng. people manager if a dedicated resource can't be found or hasn't worked out. At least their job is already interrupt driven and focused on knocking down blockers for their team.