The beauty of generic questions is also their Achilles heel - they are generic.
In some cases, a generic Q/A serves as catch-all for what is repeatedly asked, especially by new users that don’t research before posting. We all know the “I left [perishable item] in [someplace warmer than a fridge] for [random time]” questions and closing them as duplicate not only protects the sanity of the more involved users but also - and that is often overlooked - gives the asker an immediate solution without having to wait for someone to write an answer (and the community weeding out the “did the same, was ok” or otherwise not helpful ones).
On the other hand, generic Q/As tend to have long, complex answers that need to cover various perspectives. Not all askers are willing to read through a lot of text when they are looking for a simple answer. The nice thing to do when closing especially a new user’s question as a duplicate of a generic Q/A would be to add a comment summing up the crucial bit, but as this isn’t required, it doesn’t always happen.
On the site, we follow the philosophy that we take real problems and suggest solutions tailored for the specific asker. Trying to come up with a generic Q/A whenever we encounter duplicates would contradict that. The existing system of handling duplicates is sufficient for most cases and has actually a few additional benefits - users e.g. tend to use different search terms and some variety in phrasing can help them to find one of the existing Q/As, which are then linked to more.
So in short, of course you are free to post a brand-free question that may get good and complex answers (or self-answer), but there’s no real need.
And yes, whether two questions are duplicates often leaves some room for debate (which can be sometimes seen in the comments), but that’s where the community voting happens. Posts get closed by the community and occasionally reopened after a discussion. You are free to vote as you see fit.