What types of problems could lead to 10 logic board failures across 3 different laptops?
I don't believe anyone can give a definitive answer. I would guess the Apple Tech Support folks would have indicated something by this time.

I personnaly have never heard of this level of repeated failures for a single person and/or device. But I have seen low levels of failures of Apple devices at a much lower, single instance level many times over the years. I have seen that I believe to be failure caused by "power spikes" on both the AC power ( very limited for Laptops ) and but more often thru other connected devices which the the like cause of the failure. Connections thru USB, ethernet - RJ45 cables can cause "logic board failure"/"logic board replacement" situations.

I would suspect it is not your computer itself but the ecosystem of things your device is connecting to that a leading to this level of repeated failure that is hard to figure out ( apparently ).

Can you paint the picture of your ecosystem of devices and home(s) of this device over the time period of failures?