This is proved on pdf 72-73 here and invokes the Schneider-Lang Theorem (found earlier in the same PDF) to close it out. The source linked is Murty & Rath, Transcendental Numbers, 2013.

Tao cites the 1906 paper here so maybe he can furnish a copy. Gordon (1958) shows PNT implies the desired conclusion here by the very end of the paper: but he similarly mentions the Landau (1906) paper. At the least, 1958 < 1963, so this constitutes a slight lowering of the upper bound!

You may wish to consult the paper (SciHub) entitled:$$ $$Krause, E.F. (1996). Maximizing the product of summands; minimizing the sum of factors. Mathematics Magazine, 69(4), 270-278.

(The Eisermann paper linked proposes the [existence part of the] FTA has proofs in "three families" ...)

@JoelDavidHamkins I'm not sure if this would be illuminating in the sense of, if you play the game with spheres around a sphere, the reflecting strategy works – but if neither player notices it, then you can collect a lot of data for naught. (Yesterday, A Glazyrin suggested the sphere game but whenever you place a sphere its reflected sphere is also placed; so, each move involves two spheres. Another intriguing game!) $$ $$ To answer your question directly: I have the vague sense from messing around that player one can "symmetry break" in some way so as to win, but I don't trust my intuition.

is there a reason to omit $a$ and $b$ from the question (other than the original motivation)?

The link is not opening for me; I suspect it has been migrated to: empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf

If I had to guess who to ask: Gregory H. Moore, whose history around forcing connected to an MO question I asked some years ago.

This may violate one (or more) of the intended exclusions, so I leave it as a comment [for now]: How about the implementation of Shor's algorithm where $\zeta(2) = 6/\pi^2$ arises as the probability two integers are relatively prime? I wrote up the simplest explanation that I could for this value in MO 454427, which asks about proving that the probability is nonzero. Perhaps this value, or some interpretation of its use, satisfies the whimsical motivation!

You can see many proofs of 1+...+n = n(n+1)/2 here. Elsewhere on MO, I have several proofs that n^2 - n is even; see here.