“NO” and thus the reason for the ‘grandfather clause’ in A2S1C5:
“...or a citizen at the time of the adoption of the Constitution.”
If the framers had intended for dual citizenship, they would have not included this and thus the reason for the War of 1812, to force the Brits to discontinue their hold on the new American citizens that were former Brit subjects. This however did not completely stop the Brits and so in 1868 it was formalized by the Expatriation Act of 1868.
There is also the fact of the distinct difference in citizenship qualifications for Congress & President and for that you can go to the Federalist papers & the Anti-Federalist papers for clarification. Congress has 535 members, The Executive only 2, thus the reason for more stringent qualifications for the Executive who is also the commander in chief of the US military.
That is an excellent point, and well worth repeating.