It's a legitimate legal question which can't be clearly answered from existing law.
The British law to which proponents of citizenship look to is statutory, not common law, and there was no like statute in the US when the Constitution was adopted. Based on the common law that pre-dated the statute, it's an open question. Those who dismiss the matter as settled (in either direction) are being intellectually dishonest.
The last person to have been elected President of the United States by lying about and destroying the reputation of his opponents, in just the same way as he had been elected to lower offices previously, rather than running on a positive, well-grounded platform offered to the American people, was? Barack Hussein Obama.
When he did it this way conservatives were outraged. When Trump does it ... what?
I find Trump’s tactics loathsome. Furthermore, I question the wisdom (if not the intelligence) of people who like him because he operates in this way.
If Trump becomes the nominee, this will not end well.