The question is where the burning smell came from. Shorting wires can cause an arc and burn wire insulation or anything adjacent, wouldn’t be unexpected for that to make a smell, but popping something in the PSU is a bigger issue, yet if the PSU still seems to work, and I assume the wires were coming from the 12V rail to power the fans (?) then it’s probably fine, except if the fan power was coming from a motherboard header, then you may have damaged the transistor modulating fan speed, or a trace supplying power to the header, and while it may work now, if it fails later then it might have been damaged, like the trace delaminated off the PCB, and start oxidizing the exposed copper. The odds are low, but we don’t know exactly what happened.
Are they designed to shrug off these things? To some extent yes, but I wouldn’t make a habit of it. It’s not whether the OS was installing when you did it, it’s that the system was powered on at all, or even safer would be don’t work on it with the power switched on at the PSU rear switch. Do leave the PSU wall outlet power cable connected, as this *should* provide an earth ground to dissipate ESD, as long as you remember to touch a ground (like the metal of the case itself) before anything else internal to the system.
Also, not to be judgemental or anything, just playing devil’s advocate as we’ve all done something without thinking at some point, but if shorting wires has damaged the PSU, this shouldn’t be covered under warranty, nor seller DOA policy, so no exchange, eat the loss and buy another PSU. That is only one way of looking at it, the other way is that if properly designed, it should have shut down without any damage, but with it still working, the damage could be a longer term type before it reveals itself.
Your poll should have had a 3rd choice of cross your fingers and hope for the best. If there is something wrong with one of the PSU’s power rails, and later it fails, it still has the protection circuits and should shut itself off before harm comes to the system, unless it’s very poorly designed, but I doubt that a modern 750W Thermaltake is that bad – but you could seek reviews to see if there are any reports of protection circuit problems.