The idea of program as an interpreter of an input stream of instructions seems very intuitive to me.
On the other hand pure functional programming with all those monads IO monad hacks doesn't even seem to map to what we are trying to use computers for.
I would go as far as saying that not only functional paradigm but also Turing machines and lambda calculus are inadequate mathematical models for how we are using computers. In those computational models the computation is bounded - it converts a set of inputs to a result and then dies until somebody runs it again.
Most modern applications of computers are unbounded computations. The machine reads an input stream (eg. keystrokes and mose clicks) and produces another stream od outputs (such as instructions to modify pixels on a screen).
In some sense that is true. The idea that most programs are just layered interpreters has been explored in: http://degoes.net/articles/modern-fp