Why does Dave Ramsey give some advice which makes no mathematical sense?

Some of the things he says I agree with, but three points really stick out to me.

1. He pushes index funds over ETF's. Statistically speaking an ETF is going to almost surely be better long-term than an actively managed index fund.
2. He is against credit cards. I NEVER fall behind on my CC's, so why shouldn't I take advantage of the free money from it? It is free $ if I don't fall behind.
3. He says if you have cash to pay off a house, even if it is like at a 3% mortgage rate. Why would I not invest that $ in the stock market which will almost surely return several times that amount?

Why does Dave Ramsey give some advice which makes no mathematical sense?
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