The turing machine is a mathematical concept to describe computability and complexity. Although we often talk about a head moving across some infinite tape, this is just to aid thinking and understanding. You don't need that. The TM's formal definition only needs basic discrete mathematics.
Alan Turing wasn't interested in building a computer or some mechanical machine. He was interested in determining exactly what an algorithm is. He started by reflecting what an appropriately educated human being could compute given a pen and infinite piece of paper.
According to the Church-Turing thesis everything that can be computed can (also) be computed by a turing machine. Thus every computer program (which by definition is computable) can be executed by a suitable turing machine. In this sense, it could be considered a model for software, but a very weak one. Formally giving a turing machine that checks whether a string is a palindrome is tedious. Writing a Ruby or Python script to do that is a few lines.