Sadly, there is no standard. The core is one of your CPU temps. The "CPU temp" is probably that of the shield (the CPU's out case).
You cannot assume anything is still working properly just because it used to work properly. If you think about that, it does not make sense. Cars suddenly break down without notice. My coffee pot suddenly stopped working. The air conditioner was working great, then stopped working. As for power supplies, they are constantly getting beat up by power anomalies like surges and spikes, dips (opposite of spikes) and sags (opposite of surges). The only way for "normal" users to ensure their PSU is good, is to swap in another known good one and see if the same issues remain.
You don't have to replace TIM (thermal interface material) just because years have passed or even if it dries out. Remember, the purpose of TIM is to fill the microscopic pits and valleys in the mating surfaces to push the insulating air out. Even if the liquid portion of the TIM (which is there just to make the stuff flow out of the tube easier and make it sticky) dries, the solid materials are left behind, taking up the space and keeping the air out.
BUT you must ALWAYS replace the TIM once the cured bond has been broken and I fear that may have happened when the mounting screws came loose. So I would unplug from the wall, remove the heatsink fan assembly, clean both surfaces thoroughly and apply a fresh new layer.
That will happen if the BIOS PC Health threshold setting is set at a lower temp as a safety measure.