(To answer your parenthetical question at the end: It is indeed possible, already for surfaces. I don't remember the construction though - maybe it's something like remove a divisor from a Hirzebruch surface? In any case, these examples shouldn't work here, because if $Z$ is open in its closure, then it will contain a $\mathbb{P}^1$ and so can't be affine. On the other hand, if $Z$ is not open in its closure, then the question is a bit ambiguous because there is no natural scheme structure on $Z$.)

Do you have a source for this speculation by Scholze? It's very interesting & I hadn't heard of it before.

In my opinion, this question would clearly be on-topic here, if it weren't for that Konstantinos's answer in Chris's link also answers this question. It is clearly not trivial with $5$ replaced with an arbitrary integer, evidenced by the fact that no such proof has appeared in the linked question (despite having 60 upvotes) and I personally consider the $m=5$ proof there also quite non-trivial. The question also demonstrates awareness of relevant literature, and I would say is of general interest to mathematicians (or at least to me)... but it is indeed arguably a duplicate.

@ToddTrimble If the infimum of $q^2|x-\frac{p}{q}|$ over all rational approximations $\frac{p}{q}$ with $q$ odd is zero, then this sum is unbounded. (For each fixed integer $k$, consider $\displaystyle\sum_{i=m}^{m+kq-1}(-1)^{\lfloor ix\rfloor}$ for large $q$ and "generic" $m$.) Almost every $x$ (in the sense of Lebesgue measure) satisfies this - this follows from the Duffin-Schaeffer conjecture (now the Koukoulopoulos-Maynard theorem), but I'm sure that's massive overkill.

@Libli: Now that Friedrich's Knop's answer has appeared, feel free to unaccept mine and accept his instead. Interestingly, the Andreev-Vinberg-Èlašvili argument also does this split into semi-simple and unipotent, but then they cleverly (almost magically) apply the classification of invariant inner forms on $\mathfrak{g}$... it seems plausible that my long calculations are somehow encoding their more conceptual Lie-theoretic argument.

My experience: Research requires infinitely more resilience (and this cannot be overstated), but there is basically no mathematical difference between how I approach research and how I approached olympiads. But I learned deeper math and did olympiads at the same time and each influenced (positively) how I approached the other. I imagine if I had focused on one exclusively the patterns of thought would not translate as well. It seems many others found it helpful to alter their mathematical personalities for research.

Very nice indeed. By the way, should the relation $L_ig_{i+1}=g_i$ instead be $L_ig_{i+1}=g_iL_i$ (in your last paragraph)? Also, does this argument also prove triviality for $r=3$ and general $m$? It seems like it should to me, but maybe I am missing some obstruction.

@RobertBryant: Upon closer inspection, it appears that the paper may be assuming $m\geq 4$. In any case, the $m=3$ case is not needed for Voisin's argument.

@LSpice It is assumed in the paper that the codimension of this subspace is more than $2(2m-2).$ (By duality really the only relevant condition is that the codimension is $\geq 3$ (speaking of, I believe in this question it should be dimension $\geq 3$, not $>3$, no?)). Generic here means generic (in the Zariski sense) among subspaces of that fixed dimension.

For obstructions to infectivity, look up the generalized Bloch conjecture. For obstructions to surjectivity @Qfwfq's comment should be the only rational obstruction; there are other failures of surjectivity known (look up papers giving counterexamples to the integral Hodge conjecture). At the end of the day, assuming Hodge and generalized Bloch, there is a complete characterization of when injectivity/surjectivity fails rationally.

Yep - pick a primitive ample class $H$ and $r-1$ other divisors $E_1,\cdots E_{r-1}$. Then take $D_i=E_i+nH$ and $E_r=H$ for sufficiently large $n$.

This follows from uniruledness being a closed condition - this is proven e.g. somewhere in Kollar's book on rational curves.