Spurious as usual. The 'requirement' for transverse is that is the very nature of such GR far-field GW solutions. You are the only one suggesting otherwise.
First, the "very nature" is not a category of GR. Second, who cares if I'm the only one? Do you want to suggest that majority opinion is somehow important? In this case, one can ignore your #1 simply for being in disagreement with the majority opinion. You are the only one who proposes this argument.
Third, in this first point I do not care at all if the TT gauge has some problems or not. The majority opinion (which I, just to clarify this, support) is that the TT gauge is fine for GWs. But, for the sake of the argument, I do not defend in point (1) that the TT gauge is fine for GWs, but assume that there may be some problems. In this case, to use the TT gauge would be wrong. And, therefore, the position of the majority would be wrong too. Nonetheless, this would be only a problem of the TT gauge, not a problem of GR. And even less a logical problem of GR.
See above. In far-field there is only transverse wave. That's the physical 'fact' of GR's GW's.
No. There are valid GR solutions which violate the TT gauge and have longitudinal components. The TT gauge is a gauge condition. It is not a "physical fact", it is not even a theoretical fact of GR.
Again (and you have failed several times to respond already) - cite some recognized authority claiming longitudinal wave component of a physically generated GR GW exists in far-field. Well?
Why do you think such sources should exist?
You obviously don't get the point that I accept here, only
for the sake of the argument, a hypothesis which is, in reality, wrong, namely that there are some problems with the TT gauge for GWs. In reality, there are no such problems, so there is no reason for standard GR sources to consider them.
Both water wave properties are totally irrelevant to OP scenario - globally axially symmetric far-field GW's. Why keep this pointless merry-go-round diversion up?
They are not, because the math of the water waves is the same as the math of 2+1 dimensional gravity waves, and is globally axially symmetric. And from a mathematical point of view 2+1 dimensional gravity is sufficient to show your errors. Given that you have not responded to all my attempts to obtain precise purely mathematical information, I think that it may be helpful to use examples of 2+1 dimensional gravity waves which have a simple model in form of a water surface.
I have no problem to modify the spherical model a little bit so that it will be Euclidean in the far field. Use an undisturbed surface around a mountain described by a high $h_0=(1+r^2)^{-1}$ and then add surface waves to it. Far away from the center $h\approx 0$, thus, the far field is Euclidean (Minkowski if we add time). Note that in this case, with waves of type $h=h_0+\sin(r-ct)$, the waves are exactly transversal in the coordinate $r$. But this coordinate $r$ has nothing to do with the physical radius, which has to take into account that we have ups and downs on the surface. So, the TT gauge would favor the unphysical coordinate $r$, which does not define the true, physical radius.
Try imagining being serious and actually dealing with central issue in OP. Global geometry reveals impossibility of GR's transverse shear strain GW's. The point of then arguing over some hypothetical nuances that assume such impossible waves somehow exist, in your imagination, is worse than stupid.
Sorry, but this is even lower than paddoboy- level polemics. With such cheap attempts to use polemics to get rid of simple mathematical arguments you will not win any scientific discussion.