Most mathematical proofs translate quite badly to formal languages. Their formulation often relies strongly on the peculiarities of language.

Consider the "it's easy to see that" from the article. There is no translation of that into a formal language beyond either introducing the claim as an axiom, or actually providing the proof. It also means that every usage of a well known theorem needs to be explicitly mentioned.

The end result would be an incredibly verbose proofs. Moreover, the verbosity of the proof wouldn't relate very well to the complexity of the idea behind the proof. This might bloat rather simple or elegant proofs, and us mathematicians really like our elegant proofs.

Put another way, this would require writing down a lot more than the epiphany that lead to the proof.

In general, it might be nice to somehow make proofs more tree-like rather than linear. That is, show the proof from the highest level, and allow a reader to 'expand' the claims to see their proofs. Down to the 'leaves' which are considered obvious.

You would be trying to reverse a couple millenia of mathematical tradition if you wanted to convince all mathematicians to only write machine-readable mathematics. As appealing as the idea may sound to computer people, mathematicians would find it quite repugnant if they were no longer "allowed" to write mathematics in a casual style (as if it were even possible to forbid them from doing so).

That would also require re-doing all of mathematics in a constructive setting, which is a great idea as far as constructivists are concerned but it has both political and technical opposition. It's also the case that a proof in Coq doesn't have to be enlightening, especially if you discharge it with some inscrutable hammer-like custom-written pile of LTAC.