So is the issue raised by Misner actually correct?

Feel free to believe otherwise. I have explained and justified my decision.

Please point to any equation(s) and/or steps leading to such in rebuttal by Yilmaz et al, that iyo is obviously false. Also note that both parties accuse the other of supporting a theory that does not match Newtonian gravity in the weak field limit.

So, we have one claim of mathematical invalidity of a claim (Newtonian limit of GR) which is part of standard textbooks, vs. a claim of mathematical invalidity of an alternative theory nobody has really cared about. The first claim is extraordinary, thus, needs extraordinary evidence. I have not seen it. The second claim is nothing extraordinary.

And it's not an obvious discrepancy like Keplerian orbits of a test mass, but a more subtle one involving role of certain stress-energy-momentum tensor components.

It is sufficiently obvious. There is a difference in the equations. This may show up already in the Newtonian limit, or not. Once the Yilmaz side claims that GR has the wrong Newtonian limit, but not his theory, there is not even disagreement that it shows up. If it shows up, and if the GR Newtonian limit is correct, Yilmaz theory is dead.

So, it obviously reduces to the claim that textbook-level mathematics - the Newtonian limit of GR - is invalid. This is an extraordinary, crank-suspect claim. Given that the conflict has been already evaluated by the peer-reviewers, and the claim was rejected, I do not plan to second-guess.

What constitutes a bad solution in this case - one that is genuinely intrinsic to the theory? And before you answer, recall how often you have brought up the unreasonableness of e.g. CTC's, wormholes, white holes as alloweable solutions to GR's EFE's. If you hitch your wagon to a star, better check it's not a wandering one.

One where the proposed alternative has a better solution.

If, in particular, the domain where two theories disagree is one where the prediction can be tested, the fate of the theory is decided. The Newtonian limit of GR can be tested.

If it is outside this domain, it is more problematic to show that one theory has to be rejected. One needs other arguments - like that one theory violates other well-known, well-established principles, the other not.

And if there is not even an alternative theory which does not have this problem, then the theory with the unreasonable solution will be used as the best what is available today. That means, one recognizes that it is false, but a false theory which is a good approximation is better than nothing.

Solutions with causal loops and nontrivial topology I reject based on another, ether, interpretation of the Einstein equations and my ether theory as alternatives which do not have the problem.