What benefits are obtained by allowing the occurrence of free variables and open terms in lambda calculus?
A subterm of a closed term is not necessarily a closed term. A calculus of closed terms would have to model open terms as well anyway. Pretty much any definition on terms relies on induction over the structure of the terms, and to define something for closed terms of the form $\lambda x. M$, you need to define that thing for the open term $M$.
There are plenty of models of computation. What's the benefit of using combinator calculus when you can use Turing machines or Conway's game of life? Combinator calculus is a natural model of function application, but doesn't capture function definition. The lambda calculus, with variable names, models function definitions with variable names, which a core concept of most programming languages. For this reason, it's a more useful model of programming languages than combinator calculus.
The point of a theory is not just to be internally simple, it's also to be useful for reasoning.