the logical thing to do is track the voltage rails or like someone suggested, track usual problems on your pcb
i prefer always first one, is the thing to do
if you could get some schematic or some power rails voltages, start here then after identifying the voltage rail with problems, then atttack your regulators/capacitors or whatever
it's about learning also the right approach, not searching needles in haystacks. i repaired laptops intensively for almost 2 years at pcb level, never used a esr tester directly onboard, usually you got shorted caps, not degraded esr causing regulator to fail. usually == almost every time...
esr tester is good for smps repair (not in circuit) and for sorting your drawers or in r&d