Not that I can refer you to a specific video, but I've seen videos of repairmen dealing with circuit boards, and what they tend to do, is to place the prongs of the multimeter around the chip, to see if any voltage passes (to know if the chip is still functional), or to place them at the beginning of the chip and beginning of the where the current should originates from, to see if the voltage travels right up to the chip (no breakages in between circuits etc).
However, without a specific blueprint for the voltages that should be across the board, we can't just do that haphazardly.
Or course, you could do a simple compare and contrast with a working board, however I'm not certain you want to put the working board at risk
(from you know, not knowing what you're doing )
From what I understood, it's not the save that you value, but the cart itself.
Provided it's only the chip that's dead, and not the circuits on the board (or some other factor),
someone with the appropriate tools can try to transplant a working chip onto it,
but it's hard to save it'll work, and the cost and time taken will probably be high.
Even if it's not you, ultimately it would have to be determined if the chip is dead or the circuit board is functional,
before any decisions of what can or cannot be done be proceeded with.