I am suspicious you are chasing a ghost related to voltage/current. The main difference between the peltiers of different designation (for the same footprint) is their thickness as far as I can tell, and this effects all of their parameters.
I got marginal improvements from using digikey modules, and marginal improvements from using premium thermal paste, leading to a very small improvement after alot of effort, for the same class of device, but I was always measuring current and voltage (power)
If you still have the broke one, measure it with a caliper to see thickness compared to the one you installed and the ones that have performance problems.
IMO the problem you have with grease must be related to pressure, basically you gotta be willing to crack a module or two to figure out what too much pressure is when it has a thick thermal grease. This is the same, there is some region where you get nominal performance, and beyond that you might squeeze a tiny bit more of it by putting it under high pressure at the risk of cracking it, but if you don't get to that nominal point (with thick grease that is difficult to 'feel') then it will have decreased performance
Now, if there is substantial impedance variation between the modules, and someone set the device power supply to some resistance they got at 100,000 quantity of modules, I can see how tolerance on the module impedance can cause you a problem from another batch. The performance of the device really depends on current flow. When they mean a 12V module, its nominal, and they probobly expect a trim to get your desired nominal operating current.
I suspect that any module of the same designation will work, if you can adjust the power supply a little, to get the same nominal current, and when you use as 12.0v rail, you are totally at the mercy of the manufacturing tolerance and you are left with trying to match the device to the rail, rather then matching the device to the expected trimmable power supply.
And the same thing when you adjust current, you get to a nominal point where the system is reasonable, and then if you increase current more, you get a small increase in the cold properties of the cell at the expense of alot of extra heat and inefficiency.
Think about LED's (diode), they get very non-linear, and running them based on a voltage is not going to get the most performance out of them, since you end up with a diode curve, and really you have to adjust the voltage (and so current) to get the most out of each LED. And LED are highly efficient devices, unlike peltiers.
This might be useful to know if it breaks again, so instead of trying to match more modules, you can give a voltage adjustment (with current measurement) a try. And measuring the current use of this module (if its accessible) might help you get a 'specification' for future repairs, so you know what current you should expect.
I realized when I search this thread for the word 'amps' what the problem probably was
The thing I am curious about now, does that power supply have an adjustment potentiometer on it for voltage? I wonder if that is why these devices have shit reviews, since some of them might be getting under-powered if their impedance is higher and not cooling. Or maybe they just get the modules sorted to work at exactly 12v to get rid of the adjustment circuit