Alright, say you program a computer to verify a hairy proof (one example is the four-colour theorem that was proved in 1970s in this way). How do you know the computer works correctly? How do you know the verifier program works correctly? How do you know you encoded the proof requirements and all prerequisites and lemmas correctly?

People have been working on automated proof verification ever since the computer was invented. It is going somewhere, but the validity of such proofs is questionable. You have to accept that a proof can be considered verified by an agent, which encompasses both humans and machines. Machines do not have independent thinking; they are programmed by people, who may and will make errors in the programming. Even if you verify the proof with different machines programmed by different people, they may still all fall prey to the same oversight, and the computers will happily churn the incorrect answer.

Also, the first step in your insufficiency proof is questionable. True, many proofs can be dressed up as linear sequences of statements, expressions, and conditionals, but that's where the analogy ends. More correctly, at the very abstract level a proof is a sequence of transformations from given statements to desired statements. There is no computational state involved, nor any sense of input or output. Rather the prerequisites of the proof are the input and the results the output, and you, the verifier, are the actual program.