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I'm not a pump experts by any means but here's why I'm guessing a variable displacement pump would fall flat....

1) An AC compressor is 'pumping' a very different medium than hot oil.

2) If the AC compressor fails, the engine doesn't blow up. I doubt a variable displacement pump would be as reliable as a simple rotor type pump.

The AC pump is a piston type pump and can pump any sort of medium. That it can compress a gas means that they're built to very tight tolerances and clearances. Hot oil is easier to pump in comparison. The same style of pump is used in hydrostatic transmissions for tractors, lawn mowers, etc. They're typically pretty reliable, especially in larger ag equipment.

Very true, an AC pump doesn't kill the engine when it fails (usually). The additional complexity could be a drawback, but likely not a huge one. The cost would be a definitive drawback and the gains are likely small for a highway vehicle. I'm not advocating that people start building and using variable displacement pumps by any means - it just occurred to me as I was thinking about the constraints on the oiling system and the drawbacks of what's commonly used.

There's typically multiple pistons/cylinders in a variable displacement AC pump which would give redundancy. There's also ways to limit the min volume, so if the 'variable' part fails, it would still be set to deliver a minimum of oil.

A simple rotor pump can lose the relief valve/spring or get jammed by gunk and fail catastrophically too.