Ahhh, yeah got you
Let me point you to two famous really famous examples of this, the first is the one from the the Odyssey. Odysseys is the king of Ithaca (and just a gernal bag of selfish garbage but whatever) after a few decades of odyessing he comes home to his wives house full of suitors trying to take the throne, he ends up proving that he's alive and well and ready to take back his kingdom by stringing a bow so strong no other man could do it then shooting an arrow through 12 axe heads. He uses this impossible feat of strength to show his own might and thus his worthiness and reclaim his throne (then slaughter everyone) So you could do something similar in your story, as in having a seemingly impossible test and have your character do something to show their strength.
The second even more famous example is the sword in the stone, which is just what it sounds like, a sword stuck in a stone and anyone who can pull it out is the rightful king of England. So a bunch of jacked guys try forever to use their odysseus like strength to pull it out but no dice. Then this one nobody comes and instead of pulling on it like mad man just calmly picks it up and pulls it out, and this nobody is king Arthur. That's where the whole "a gentle hand will rule the land" comes from, in this one instead of just being strong the king needs to be smart and gentle. You can use a similar test of intelligence or compassion in your story.
Which one you should use I think depends on your character and also the kingdom they rule, if your world relies on a strong ruthless and warrior king then go for a physical test but if you're aiming to show that your idea of a good ruler is more thoughtful and peace aimed then I'd follow suit as to the second.
And I mean what specific test you want to use is up to you because you're the only one who can tailor it to show the real character of your character. Just like how Odysseus' test reflected his character of being a war oriented alpha male type (Which was something highly valued in his culture) Arthur's test proves that he'll become a fair and nobel king (which is something the culture that created him wanted in their own idea of a perfect king)
SO uh, yeah, brain v brawn, your test will reflect on your character