Will turning a device on and off via power supply hurt it?
The risk of damage depends on the individual design of the electronics, and can't be generally asserted.

I'd argue that consumer electronics should be expected to be relatively indifferent to sudden breakaway of supply current. Experience tells us that some devices are resilient, others not, for often very different reasons. 

For example, JRE mentions that hard-drive containing systems might suffer. I think that might be an early 1990s phenomenon and mechanical damage to hard drives due to loss of supply would be a thing of the past. Still, hard drives contain data, and writing half of something will definitely break that something. It's a software problem to solve that issue (resilient file systems and broken file handling).

For inductive loads, a sudden change in current (e.g. due to a switch being flipped, moreso a relay being opened, a circuit breaker firing) will lead to a voltage spike. That's just physics. Whether or not that spike is dangerous to device itself, or others on the same phase, is up to the exact moment that breaking happened, the inductance, protective measurements on both the inductive load as well as other devices and sheer luck.

So, no general statement can be given. I'd argue that switch-mode power supplies these days are both relatively well protected against voltage dropouts and surges, and relatively low in inductivity. Noone but someone with time and a test stand can tell you how often you can externally switch your amplifier off and on without it taking damage – be it billions, thousands or just dozens of times, and the device manufacturer will of course never guarantee anything but switching off the device itself before removing it from supply. 

I'm sorry, but you're on your own with this, and it's a basic task of personal risk assessment. I wouldn't worry too much if the cost of the devices in question are, divided by the amount of cycles you are pretty certain won't destroy them, is lower than let's say electricity costs of letting them run through.