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It's a nice exercise but doesn't this ultimately proof that using these formalized proofs is flawed? That this system can be made to 'proof' anything, as long as the premises are logical by themself?

Or am I reading too much in the mathematical term 'proof'? Because it sounds more like 'verifiable reasoning' to me.

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You already hit the important part with "as long as the premises are logical". Since the premises here are flawed, the proof falls apart. It isn't a flaw in formalized proofs. Getting flawless premises is the challenge for trying to use this for philosophical proof - the concepts are typically not as well defined as, say, what you see in geometry class.

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