Hello,
If the power supply is too weak, it will of course fail first. A question of luck or lack of it (read: protections in the power supply or their absence), whether it will burn the rest of the devices connected to it or not.
Symptoms of a weak power supply: the voltages measured with the meter differ by more than 5% from the reference voltages (i.e. theoretically at 12V is allowed 11.4V to 12.6V, and at 5V from 4.75V to 5.25V). The second symptom may be that the voltages are unstable - e.g. the meter while charging the system will show everything from 11.90 to 12.50V on the 12V rail - although it is within the tolerable range, it is not a good symptom). These two symptoms most often cause computer freezes, unexpected reboots or blue screens. Third thing - the air coming out of the power supply will be very hot. It is known that it will always be warm, but if it competes with a hair dryer, it means that the elements will overheat. This can lead to the fourth terminal symptom - a thick cloud of smoke from the power supply.
If you suspect that the power supply is too weak - especially if it is not a branded power supply - it is not worth risking the life of the rest of the equipment in such a configuration.