Since you point out that sloppiness is due to poor or lack of maintenance, that seems to support the superiority of push rods. From your statements it appears that cable and pulley systems are, for whatever reason, not properly maintained , thus sloppy. Pushrod systems OTOH seem to not require such fastidiousness to stay precise.
Sloppy cable systems are a symptom of lax maintenance, and that makes one wonder what else is ignored in such airplanes. I have flown nearly-new Cessnas, and they are tight and easy and precise. As a mechanic I have found slack cables, sticking pulleys (sometimes altogether seized), worn hinges and a lot of other stuff. Once the worn parts are replaced, pulleys replaced or freed up and relubed, and the cables properly tensioned, there's a world of difference.
Pushrods? I've found their bellcrank bushings worn or sticky. Found their rod ends worn and sloppy. Found their guides loose and rattly, wearing the pushrod tubes. And worn hinges happen in those systems, too.
Cables lend themselves well to routing through difficult areas and around corners. Pushrods don't. Instead of a couple of pulleys to change direction, you need a bellcrank and the room for it. Long pushrods need intermediate supports in the form of guides or more bellcranks.
You won't find many pushrod-operated controls in high-wing airplanes. Getting pushrods out of airplanes to replace rod ends, or to inspect for defects and wear such as in the Cessna 400 aileron system inspection, involves a whole lot of disassembly.
Pushrods are fine. Nothing wrong with them if the system is well-designed. But to disparage cable systems as being inferior is mistaken, since almost all of the slop and drag there is due to a lack of maintenance in airplanes decades old. If we maintained the engines like the control systems get maintained, there would be a lot of engine failures.