A worn contact, not a software setting
A mouse switch closes a metal contact, and as that contact settles it bounces. The firmware ignores state changes for a few milliseconds afterwards — the debounce window — so the bounce never reaches your computer. As the switch wears, the bounce lasts longer than the window, and one physical press arrives as two clicks.
That is why the fault is progressive, why it gets worse under a hard click, and why no driver setting fixes it. Windows' double-click speed slider changes how the operating system interprets two clicks; it does nothing about a switch that is sending two.