A few years ago we switched to a "first-to-file" system and this is a direct consequence of that. Someone who didn't invent something can now file a patent. Doing so is a lengthy, expensive process, so the immediate consequence is there is no more "public domain" inventing.
This needs to be reversed.
Saying that first-to-file means a non-inventor can lawfully obtain a patent is also wrong. It seems to imply that a new, non-obvious invention can only be made by one person at a time. This is not true in competitive industries where many people are working to solve the same problems and will often come up with the same new solutions independently. First-to-file encourages them to apply for their patents early.
There is such a thing as "public domain" inventing, as you call it. It is simply done by (1) inventing, (2) publishing, and (3) not applying for a patent.
You do have to lie and say you thought of it independently; something impossible to disprove. The number of people who'd lie for big money is... innumerable.
You can patent ideas thousands of years old - if they're not well known to practitioners.
As for obtaining a patent by fraud, no comment.
If I've invented widgets (or think that I've invented widgets), I should be able to know if I can patent and sell them, without years later getting sued because someone once did it in their basement and left it at that.
Also, first to file has the same problem you mention. You can invent something, never publish, and get sued by somebody who reinvented it later.
Edit: this was an interesting related HN discussion - https://news.ycombinator.com/item?id=17240429
But most companies would rather just patent something.