As a female line grandchild of the monarch, was Prince Charles not entitled to be a Prince at birth even though he was second in line?

George VI issued special Letters Patent that all children of Princess Elizabeth and HRH, Philip, Duke of Edinburgh, would be titled prince or princess. Otherwise, protocol states a princess doesn't pass her status to her children, that children take on the status of their father.

George decided that since Elizabeth was the future monarch, her children all would be prince or princess. He had not made Philip a prince, just a royal duke, so the children all would have taken on the titles and status of a duke. Philip was HRH, Duke of Edinburgh, Earl of Merioneth, Baron Greenwich. If Letters Patent had not been issued, Charles, as eldest son, would have taken the title next down and been HRH,The Earl of Merioneth, and younger siblings would have been Lady or Lord.

https://www.heraldica.org/topics/britain/prince_hi...

Children of HRH The Princess Elizabeth and the Duke of Edinburgh (Oct 22, 1948)GEORGE THE SIXTH by the Grace of God of Great Britain Ireland and the British Dominions beyond the Seas King Defender of the Faith To all to whom these Presents shall come Greeting Whereas His late Majesty King George the Fifth by His Letters Patent dated the thirtieth day of November in the eighth year of His Reign did declare His Royal Pleasure that certain members of the Royal Family therein more particularly mentioned should have the style title or attribute of Royal Highness And Whereas We are desirous of defining and fixing the style and title by which the children of the marriage solemnised between Our Most dearly beloved Daughter Her Royal Highness Princess Elizabeth Alexandra Mary Duchess of Edinburgh and His Royal Highness Prince Philip Duke of Edinburgh shall be designated And Whereas for that purpose We deem it expedient that the aforesaid Letters Patent should be amended and extended la manner hereinafter declared Now Know Ye that in the exercise of Our Royal and undoubted prerogative and of our especial grace we do hereby declare Our Royal Will and Pleasure that the children of the aforesaid marriage shall have and at all times hold and enjoy the style title or attribute of Royal Highness and the titular dignity of Prince or Princess prefixed to their respective Christian names in addition to any other appellations and titles of honour which may belong to them hereafter And We do further declare Our Will and Pleasure that our Earl Marshal of England or his Deputy for the time being do cause these Our Letters or the Enrolment thereof to be recorded in our College of Arms to the end that Our Officers of Arms and all others may take due notice thereof In Witness Whereof We have caused these our Letters to be made Patent Witness Ourself at Westminster the twenty-second day of October in the twelfth year of Our Reign.

I find this Letters Patent a bit confusing, though, as it lists Philip as "Prince Philip," eventhough he was only created Duke of Edinburgh, not formally a "prince."

At the time of Charles' birth, his mother was the heir to the throne, still Princess Elizabeth. Charles, born a prince, was second in line of succession.

Huh?    Prince Charles, a Prince at birth, was never 'second in line'.    He is the first-born child, a male at that, of the late Prince Philip and Queen Elizabeth 11.   He is followed by a sister, Princess Anne, and two brothers, Princes Andrew and Edward.

All children come from the female line.

Have you ever heard of a man giving birth.