Paying down debt is setting money aside to pay for something you bought at some point in the past.
Saving is setting money aside to pay for something you might buy at some point in the future.
So, far from thinking about credit availability or credit worthiness, if you just think of debt as negative savings, you can see debt and savings mutually annihilate (like matter and antimatter), and sum to zero.
Debt is a wonderful, powerful invention -- but for many people who are unaware that debt costs interest, this simple model can help them reason about debt and decide to pay it off rather than letting it linger or taking out more debt.
Debt is tied to enforcement, often punitive enforcement.
There is an element of freedom buried in these that I am trying to suss out. I am reluctant to use evocative words to describe what lies beneath the obligations of debt, but I sense there is a qualitative difference under the surface.
If I had everything invested and nothing saved in bank account, I might get into situation when I need to liquidate my investments ASAP, which could be very unfortunate, timing-wise.