It’s probably an advantage. You want your mouth to hurt so you stop there rather than damaging something more internal like your throat or stomach.

I'm sure the hippo wouldn't eat burning hot watermelons.

When we stopped eating twigs in favor of nuts and berries.

Note: this is an entirely made up fact.

The answer is almost certainly because of how much brain we're trying to fit in our heads.

Evolution has already given us hugely weakened jaw muscles, shrinking jaws, painful birth, years of helpless infancy and chronic sinus problems, all in the singular pursuit of MORE ROOM FOR BRAIN.

To be honest I'd say all of those are higher up on the priority list than 'thicker mouth skin for when I eat a thing too hot too quickly once every couple years'.

Which animal ever eats something too hot that burns their mouth? How do you know they don't burn their mouths in the same way? Do you think hippos use the top of their mouth or their teeth to crush watermelons? Is your fragile mouth the only thing stopping you from crushing entire watermelons?

Does the thickness skin impact it's ability to feel pain?

It doesn't. Burn a slip of paper with a lighter. Now burn an orange. Notice how the the outer layer reacts to the fire?

Your nerve endings also exist in the outer most layer (your mouth vs your deeper flesh)

Are you a mammal? Congratulations, you have TRPV1 receptors in your mucus membranes (avian supremacy ftw). They exist to keep potato brain mammals from eating hot things. These will fuck up your insides. Apparently enough of a problem that mammals needed them. Fun fact: this is enough of a problem that ancestral peppers evolved a means to exploit it and exclude mammals.

But off topic. There's no quantity of flesh that will make your mouth heat proof. Turns out first degree burns still hurt like fuck.

@doomsdayrs probably, to regulate brain temperature. There is a theory on yawning that its purpose may be brain cooling. So, the roof is basically a thermal interface.

Your body telling you something dumdum. ;)

What other animal eats hot food? Has it ever been something to select for?

Does the thinness of the roof of one's mouth keep them from propagating their thin roof genes?

Has your ability to procreate or the survival of your offspring ever been threatened by burning the roof of your mouth?

There is your answer.