So why doesn't everyone fill in whatever gives them the highest chance of getting into college?

Because by and large most people are basically honest.

Same reason everyone doesn’t just walk out of restaurants without paying.

But there have been high profile cases where it has come to light later and people have endured some social shame.

That's not what I meant, let me rephrase my question:

How does someone know what they should fill in?

Is it fine to fill in African American if one of your grandparents was black? Or only if your skin is dark enough?

Hard to say. General consensus seems to be “appearance” but others go by the one drop rule.