Ramsey's belief is that all debt is bad. I tend to be that way myself. But I certainly understand why people don't rush to pay down a mortgage, as it's just about the only leveraged asset they have access to. It does presume home values will rise and investments will grow more than the mortgage interest rate after factoring in the tax deduction - but that's not a stupid stand to take even if things don't work out. There are no guarantees.
I don't like being in debt on cars so we don't borrow for those. Even if they dangle a 0% interest offer, I still have to worry about the car being upside down in case of an accident.
We worked hard to pay off our mortgage a few years early and put that extra money into savings. Being totally out of debt means that besides food and utilities, the only other money we HAVE to spend is for property taxes.
One of the defining moments in my anti-debt opinion was when I was a kid and my parents borrowed to get new windows in our lower-middle class rowhouse. I don't remember the numbers exactly, but let's say they borrowed $3,000. Pay back over 5 years. They made monthly payments of around $90 for 3 years (sum of about $3,200 paid). Then they got a small inheritance from an aunt and wanted to pay off what they assumed was a small remaining balance. They were told the remaining balance was around $2,000. On a $3,000 loan they paid on for 3 years already!
Turned out the scammy window company finance arm used a method of computing finance charge (precomputed and add) that meant there was no benefit to paying the loan early. The total they owed was about $5300 (including finance charge) and they owed that amount even if they had paid the loan off in the first month. That really had an impact on me - if I borrow money, it's on the lender's terms even if they are unfair to the extreme.
Beyond the financial benefits of being debt-free, there is the aspect of freedom. You have more freedom to make big life choices when you don't have to worry about paying back debt to make them work.