No, the exception clause or the grandfather clause had to be there because the Framers themselves were not natural born citizens. Why? There was no United States when THEY were born. But for those born after the Constitution was ratified, all they had to do was BE born. Yet without the grandfather clause, the country could not have had a president at all — until a child born after the Constitution was ratified had attained the age of 35. Long time without a president for a new country, no? That’s the reason for the grandfather clause. Nobody alive at the ratification of the Constitution (except a baby 2 minutes old) could possibly be president unless the living people in the country were exempted from having been born in the United States. Remember that there WAS no United States when George Washington was born.
Your “explanation” is a nonsequitur.