You continue to miss the real point. Non-linearity per se does not imply 'self-gravitating'. Zero Ricci curvature implies absence of *all* sources of gravitation in such region, not just matter density. Hence, explicitly, form of exterior Schwarzschild metric forbids existence of field self-gravitation.

Correct, I completely miss any point here. No physicist really cares about some popular descriptions for laymen use the phrase "self-gravitating". I would expect that many GR textbooks do not use this phrase at all, and if they use it, I would guess they use it simply to refer to non-linearity of the field equations.

Many try and impose it via 'tricks' so as to justify BH 'self-sustaining fossil field' etc. but such cannot be true to the basic statement of EFE's. Shuffling lhs 'effect' terms to rhs and pretending such legitimately becomes 'source' is often done but that doesn't make it legitimate practice.

I don't even see any reason to try. Which terms of an equation are written on the left side and which on the right side is completely irrelevant, it does not matter at all. You can write the Einstein equations as $$ G_{mn}=T_{mn}$$ as well as $$ G_{mn}-T_{mn}=0$$, you can subdivide $$ G_{mn}$$ into, say, a second order term $$ \square g_{mn}$$, which is, for small modifications, almost linear, and all other terms which are non-linear even on the small scale, and all this is completely legitimate, because it does not change the equations even a single bit. And all this plays, at best, a pedagogical role, and is otherwise completely irrelevant, and, in particular, cannot cause any foundational problem.

Similarly a hypothetical TT GW is in GR a nonsense for the same basic reason - zero Ricci curvature directly implies zero stress-energy-momentum density in such 'wave'. In other words, as Eddington maintained, all you can have is 'ripples in the coordinates'.

No. To reduce a solution to "ripples in the coordinates" requires $$R^k_{lmn}=0$$. That the Ricci tensor $$R_{mn}=0$$ is necessary for this, but not sufficient. In particular, you cannot transform the Schwarzschild solution into the gravitational vacuum (the Minkowski metric) by a simple transformation of coordinates.

In that respect I no longer claim absence of GW's is specifically evidence against GR.

Which would be anyway nonsensical, because the GWs which would have to exist, if all what we think we know about stars, galaxies, and what can happen to them would be correct, are below the threshold we are able to measure. Thus, observation of GWs would be the unexpected things, pointing to new, unexpected physics of sources of unexpectedly strong gravitational radiation. (At least this was correct a few years ago when I have checked, if you have different, more actual information, tell me about this).

While in Yilmaz theory non-zero GW energy-momentum has a self-consistency entirely lacking in GR, TT character could not be exact owing to the 'self-gravitating' requirement. For other reasons not to be divulged here, even such Yilmaz almost purely TT GW's are not possible.

Sorry, but claims that GR is not self-consistent are extremely dubious, a clear indication of crank science. So, if one simply ignores those who make such claims one usually does not make an error. (An exception is Logunov's RTG. He has made the error of making wrong claims about inconsistency of GR, which has extremely harmed his scientific reputation, which has also harmed the reputation of the theory of gravity he has proposed as an alternative, but which is nonetheless a viable alternative to GR.)

GR has a lot of weak points, from singularities, the non-existence of a local energy-momentum tensor for gravity and dark matter to incompatibility with quantum theory, but this does not mean inconsistency. GR is an internally consistent theory. At least I have not found any inconsistency.

If there is some consistent formulation of Yilmaz theory I don't know. At least I have not seen one which I would have recognized as defining a consistent theory.