JJ - My reply is broken down into 2 segments. A somewhat useful segment. And a useless segment. I specialize in uselessness.
The somewhat useful segment: I agree with your direction to replace the engine and outsource the labor. If you're going to pay someone $950 to do the labor, make sure they're not just some backwood rednecks. I did that once, and it didn't work out so well.
The useless segment: I think you're letting this theory about it being rich cloud things. It generally makes no difference based on your direction though. Your direction is solid. But I don't believe even if it were running rich it would cause it to lose compression. It would appear your vehicle is a CVT, because you say it was a rental car. I'm not sure any place rents a manual transmission car in the U.S. But here's a for instance. Let's say your cam chain has seen a lot of wear, due to poor lubrication. It is my theory that Hertz never does even 1 maintenance on a car. They buy it, slam it out on the road, pull it off the road at some set miles and sell it. I would doubt they'd spend one thin dime to maintain it. Not having fresh oil in there could cause the cam chain to not get proper lubrication (along with everything else). Perhaps the cam chain could stretch enough to throw off your cam timing. Improper cam timing can lead to less than desirable, perhaps little to no compression.
There's lots of ways to lose compression. Bad oil can lead to ring wear too of course. Ring wear = low compression. I think the whole rich thing is perhaps not likely, but is beside the point in any event.
And as far as closed loop / open loop. I don't happen to know factually what a computer controlled engine does. But I'm a carburetor tuning expert. The vehicles I've had my AFR on will hardly even idle at stoich. So I have my doubts a computer controlled car shoots for stoich at idle. My LA 360 idles rich as a mofo at idle. And just get's plain *****y about idling when anywhere near stoich. I've got it idling in the neighborhood of 13.0 - 13.5 and it is still not that smooth. Cam timing and lift comes into play as well. But my LA 360 runs at about 12.5 - 13.0 at WOT and top rpm. And cruises at about 15.5. The only time it hits stoich is in transitions between power and cruising. I have to assume even modern day cars run similarly, not just aiming for 14.7:1 all the time. 14.7 wastes fuel when cruising, and could lead to burning a piston at WOT. But again, I don't know the fact as I have not applied an AFR gauge to my computer controlled vehicles.
7milesout
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View my fuel log 2020 Mirage ES 1.2 manual: 42.4 mpg (US) ... 18.0 km/L ... 5.5 L/100 km ... 51.0 mpg (Imp)