Your first fact is incorrect, a fancy word for wrong. Ike was the President, and while he's technically the commander and chief he cannot spend money, create a budget, or much else. Congress does that. Congress (back then) governed by the will of the people, and Americans made it clear to their congressmen that living under a Soviet moon was not an option. So off into space we went.
And we did. Apollo 11 was two years late thanks to the Apollo 1 fire, and delay. We could have landed much earlier.
Mostly due to political reasons. Priorities, and money went elsewhere. But our second robot on Mars is still hard at work, we've collected mineral samples from a moving asteroid, and returned them to earth, and all kinds of fun space stuff...because we can.
That should tell you all you need to know about China. For all of their PhDs, they could have been sending men into space from 1985 onward. They have issues. Mostly related to wait unti the US developes something that they can steal, and back-engineer.
Sure there was. It's called, "SCIENCE". Manned space flight is dangerous. Danger is kind of our thing.
Okay, here's the the math: Aldrin suffered from depression, which ran in his family. His mother committed suicide in 1968, and his grandfather ended his life years before. So the math in Aldrin's case was he led a rough life before and after NASA.
Armstrong wasn't a recluse, he just didn't feel it was right to cash in on Apollo 11, and the first-man-on-the-moon hype. He could have had anything, high paying media jobs, high paying consulting jobs, he could have sat as a paid member on multiple corporate boards while doing nothing but playing golf with potential clients, he could have been a congressman, senator, governor, and even President. Instead, Armstrong walked away from NASA, taught collge for a few years, and ran his small farm. Cashing in on Apollo 11 was not an option for him because he didn't go to the moon alone, and that 400,000 people had busted their butts to get Apollo 11 to the moon, and back, so cashing in would have been disrespectful of their work.
Neil Armstrong is considered a hero not because he was the first man on the moon, but because he did his job, and resisted the temptations which followed. He was a man who lived by a code of ethics. This is something sorely missing today.