I definitely hear you on the product/project distinction. It's sort of adjacent, it's sort of a subset. To me, project management is the cat-herding part of execution. And that's exactly the part where trying to get other functions to log work in the "catch-all PM system" falls apart, whether you're a project manager or a product manager juggling tickets.
You can either run it out of JIRA and only manage the work of software engineers, or you can run it out of something else and be out of sync. Probably the biggest mental trap PMs of both kinds fall into is running all status checks out of JIRA and ignoring everything about their product/project/career that isn't accomplished by pushing code to prod.
I find the people do get value out of what Product brings to the table, and absolutely buy into the vision (especially if they help define it). That doesn't make them motivated to maintain an additional project tracker for a status meeting. They also don't like flossing, even though they hate cavities.
And that's what we're talking about at the end of the day, because JIRA IS really good at what it does - either everyone gets aboard the JIRA train, or the engineers track all their tickets in something less powerful, or you ask the engineers to use JIRA AND something else.
Most established places go with the first option and then their only muscle is "ship code," many startups go with the second option until they realize they actually need JIRA after all. Then some PM (of one kind or the other) has to make the third option work so that they stop getting blindsided by the compliance work or the marketing campaign.