The author appears to have a serious misconception about Lean, which is surprising since he seems to be quite knowledgeable in the area.

Specifically, the author seems to be under the impression that Lean retains proof objects and the final proof to be checked is one massive proof object, with all definitions unfolded: "these massive terms are unnecessary, but are kept anyway" (TFA). This couldn't be further from the truth. Lean implements exactly the same optimization as the author cherishes in LCF; metaphorically, that "The steps of a proof would be performed but not recorded, like a mathematics lecturer using a small blackboard who rubs out earlier parts of proofs to make space for later ones" (quoted by blog post linked from TFA). Once a `theorem` (as opposed to a `def`) is written in Lean4, then the proof object is no longer used. This is not merely an optimization but a critical part of the language: theorems are opaque. If the proof term is not discarded (and I'm not sure it isn't), then this is only for the sake of user observability in the interactive mode; the kernel does not and cannot care what the proof object was.