The thing here is that the HDD does not have fans to cool the thing down. The power issue of running the PSU hot so to speak would more result in the PSU failing as the current being drawn places the load on the PSU not the other equipment. The other devices connected don't realise that there is an undersupply (due to PSU overload) and cut back or go into an economy mode, they either draw the current or not and if there is a 'sag' then it is either compensated momentarily by internal resistors, capasitors and alike or shut down either hard/soft.
What is the internal temp of the chasis running normal then when the PSU has been overloaded.
What is the temp of the HDD normal and with the PSU overloaded?
Where is the HDD in relation to the PSU?
Is the any fans to compensate for the increased heat issue of the PSU?
Remember its not that the HDD has to work harder but an environmental issue. The HDD was designed with Max operability at max temp. The operability will decrease as the Max temp is exceded, benchmark the HDD in both environments if you need to see more. If the HDD had to increase its workload then it must pull more amps from the PSU not the otherway round and this is already compensated for by the manufacturer.
The only waning i would give you is that if there is any type of problem when your taxing the PSU and a PCB dies while running then a spike may occur for that nanosecond but its all it takes to fry the MOBO resistors or arc for a nanosecond and fry other parts as well.