So if amendments are possible, and the Constitution can therefore grow, how is it not "living?"

The Constitution, like any other written document, has loopholes. It is the job of a neutral, rational party(the judicial branch) to interpret it and rationally judge on it - if not, people could claim that the First Amendment "right to free speech" allowed them to run into a Church in the middle of a ceremony and cuss out God and everybody in it. That's not protected speech - that's a public disruption, but if you translate the Constitution literally, it can be read that way.

As time changes, so do current situations. It's 2005, not 1787.