Electronic – Why is power of a signal equal to square of that signal

Because the mains supply is very unpredictable, and can do all sorts of things outside its nominal specification, which might damage components or at least break the nominal design assumptions. A non-isolated design also has all its voltages referenced to one of the mains conductors, which might or might not have a useful/safe relationship to other potentials in your environment (like earth/ground, for example).

If the only stuff on the low-voltage side is inaccessible electronics, then non-isolated supplies are fine - they tend to much be cheaper/simpler than isolated supplies, and lots of household equipment uses them. Even things like televisions used to work like this, if you go right back to before the time when they had external video/audio connections. The antenna connection was the only external socket, and that was capacitor-isolated.

If a human being or 3rd party piece of equipment needs to interconnect with the low-voltage side of your design, then an isolated supply both gives you a clear barrier across which dangerous voltages won't pass, even in the case of component failure, and it means your circuit is now 'floating' relative to the mains. In turn, that means you can arrange for all the electronics to operate near ground potential, with all your interconnected equipment having at least roughly the same voltage reference to work from.

TLDR: better design is not affordable.

Technical root cause is the limited speed of light. Natural root cause is the conflict of economy of scale vs redundancy requirement.

Every energy supplying element in system is protecting itself by disconnecting, stopping the supply. Disapperance of supplier, causes increase of individual load on other suppliers. Which in their turn disconnect themself. The blackout escalation behaves very much like an avalanche.

The proper design must have granularity of demand control (consumer's pool) be finer than smallest individual supplier, and total supply capacity must be greater than total demand by at least one unit of supplier's granularity. In theory the redundance of suppliers must protect from blackouts. But in reality the granularity on both sides is too coarse, and is impossible to be made finer. Because downsizing is in conflict with economy of scale. Every power plant, distribution line and other elements are built to be as large as possible and point of connection to consumer always exceeds the optimal size granularity wise. Just because of market competition it is not possible to limit the size of lines, power plants and other elements.

The solution can be found only in regulatory space. Say: add the smart load control capability to all household Energy-Star rated equipment and introduce concept of "quality of service", complex system of tariffs, uplinks, protocols, etc.