No. We have an infinitely more succinct formalism, it's Turing machines. Succinctness is not necessarily a desirable property, it just says where on the capability-tractability tradeoff something is. Turing machines can express literally anything computable, but in exchange we can't use computers to reason about them in general (Rice's Theorem). Regexes are much more limited, they famously can't even recognize HTML, but we get to automatically prove things about them.