Okay, yes, the 19th amendment did not remove the barrier against women serving as president. It didn't remove that barrier because that barrier never existed. Here's what the Constitution, Article II, has to say about the qualifications to serve as president:
Most people believe not only that the 19th Amendment permitted women the right to vote but that since women serve in Congress, the courts and other offices of government, the office of president of the United States has been de-genderized.
Not true. This important legal question exists now and has not been constitutionally addressed. The language and syntax of the 19th Amendment merely removed the barriers that prevented women from voting. It did not identify women to be qualified to become elected president.
Nice, gender-neutral language.
No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty five years, and been fourteen Years a resident within the United States.