> How hard would it be to write mathematical proofs in a modular way? And have someone check a "module" of a proof and someone else another "module" and so on?
To a large extent, this is what we do. But of course, modules by themselves don't prove a thing about what we're originally interested in so that not one module can stand by itself as any advance in the proof, that theconclusion of any module of the proof are surely needed for all the others modules to make sense and that these conclusions are more often than not formulated in a new part of language (new words, concepts, and rules to play with them) introduced in said module. Parallel work of the sort I infer you have in mind might be possible in some instances, but there is so very few of them in my opinion that it's just not worth trying.
> ...and for the bits that's possible, write in a subset of mathematical language that computer can check, or at least partially check and ask the human reviewer for feedback on the parts that can't be automatically proven.
No can't do: in mathematical writing, what is actually written is the tip of a gigantic iceberg of implied reasoning and background scenery. There might be a way to make computer understand this, but as far as I know it has yet to be found. That, and making explicit what is not in mathematical writing would make the length of any proof grow manymanyfold - and I actually mean manymany....manyfold.