Did people really commonly believe in the inevitability of the "divine right of kings"?

There is a misconception that "divine right" of kings was a long standing tradition. It's a product of state centralization in 16-17th century Europe.

The hereditary rule of kings had to be justified somehow so a legal fiction of divine right was established. As to how many people actually believed in it we can't really know, however there was pushback almost immediately, for example Republicans in English Civil War, Dutch Republic, various Italian and German states... Meaning to say It wasn't a universal concept even during the peak of its popularity.

Earlier Medieval states often operated as elective monarchies, especially those of Germanic origins. Holy Roman Empire held on to the elective monarchy from 962 to 1804. In contrast France, despite common origins, slowly moved to the "divine right" concept, and pretty much pioneered early modern absolute monarchy.

There is much more to be said for states in the rest of the world. Although monarchies, Japan and China had completely different justifications as to why the king is a legitimate king (and fall very much in the divine right category). Then were are various Native American nations with government systems which seem unusual from today's perspective.

All this is to say that while some type of monarchy was the most common system before the Industrial Revolution, it wasn't universally accepted. And even when it was it wasn't necessarily of the divine right kind.

You'd believe in the divine right of kings too if the alternative was getting hung, drawn and quartered.

it's really easy to see the status quo as being inevitable, so it might start out as an innocent enough thought, but it's easily weaponized propaganda for the people in power.

Of course they did. Until the printing press, most European societies forbade peasants learning to read or write, because educated people start getting ideas, and can exchange information in secret over long distances. The Church was the only learnin' peasants need, and The Church teaches blind obedience to authority. The King wouldn't be King if God didn't want it.

I doubt there's any hard evidence, most commoners couldnt read and would never have wrote shit like that down even if they could.

What we do have is things like plays. Shakespeare seems high-class these days, but was written for commoners. And was often critical of nobility and showed they were flawed regular people.

Anything worse than that, and you were likely to lose your head.

So I feel like the safe bet was commoners hated the royals, and just couldn't do anything about it.