Dead batteries can leech enough power to keep a laptop from starting. You can firstly try removing the dead battery and then starting it from the outlet power.
If there is sufficient electrical connection, the multimeter will beep, or else it would probably mean that power cords are faulty.
Otherwise check out these methods and if any of them works.
https://www.minitool.com/data-recovery/pc-keeps-shutting-down-without-warning.html
Please keep me updated, thank you!