Do we have to resort to intuition at the very root of our thinking?

Question

Formal first order logic is the foundation of ZFC set theory. This gives me the impression that a theoretical system has to be based on a formal logical system, with its own axioms and deduction rules. However, as I read books on formal logic, I found that many conclusions of the formal system itself are drawn through proof at the level of meta-language, e.g., proof for soundness and completeness of the system itself. This is natural, as a proof system cannot be used to prove conclusions of itself. Since meta-language is not formalized, I am considering, which one is more fundamental, intuitive thinking or formal thinking. Can we develop a formal system without resorting to intuitive thinking, or is it that we have to rely on intuition at the very root of our thinking?