Abbas Bahri's article "Five gaps in mathematics" is an interesting paper which seems directly relevant to the question. (To my understanding, all of his five objections can be resolved, but I believe this is irrelevant to the question.) twma.files.wordpress.com/2016/12/five-gaps.pdf

Just one ex...it is interesting that (depending on who you ask) Freedman's proof of the 4D Poincaré conjecture and Perelman's proof of the Thurston conjecture are incomplete/unclearly complete, as they are arguably the most widely renowned works in (resp.) geometric topology and geometric analysis. It seems that nobody has been able to reorganize the logic of the existing proofs in order to clarify their content, which seems to me to be the goal of an exposition. Also noteworthy that (depending on who you ask) the existing expositions lack clarity themselves or have their own errors, or both

The first place I'm aware of the log trick is in theorem 2 of Moser "A new proof of De Giorgi's theorem". Moser says that the proof was inspired by Bers & Nirenberg "On linear and non-linear elliptic boundary value problems in the plane", so possibly it appears there too

The estimate in #4 is due to Yau "Harmonic functions on complete Riemannian manifolds" and Cheng-Yau "Differential equations on Riemannian manifolds". Li-Yau are responsible for the parabolic analogue. By considering a harmonic function on the unit ball as a solution of a different elliptic equation on the ball model of hyperbolic space, and max principle for noncompact spaces, Yau showed that $|\nabla\log u|^2\leq \frac{c(n)}{1-|x|^2}$ if $\inf u=0$. Cheng-Yau got the version you state by using cutoff functions and the usual max principle.

Thank you, that was essentially my intuition but I failed to realize it explicitly. Could one say anything in the situation that $(M,g)$ is a flat manifold?

I can't understand any of the textbooks, so to learn hermitian and kahler geometry I had to work it out for myself. But Griffiths & Harris "Principles of Algebraic Geometry" and chapter 2 of Besse "Einstein Manifolds" are standard references

My (maybe wrong) impression is that Lohkamp's proof is not widely understood, either positively or negatively

It seems like there is a precedent of this rule being broken for notable mathematicians on major conjectures - even in cases where, for what it's worth, the papers are (at least outwardly) "crackpot", which this is not