"Is only *significant* on large scales, and in particular in the weak-field limit, no?"
Actually no, in the hard interpretation of what I'm saying -- that the acceleration predicted in cosmology is a result of assuming a negative pressure in a Robertson-Walker metric... but the universe is composed of a conglomeration of untold billions of different metrics -- there is no weak-field limit. In the softer interpretation where structure forms and "disconnects" in some way or another from the universal expansion -- basically, is modelled with something like an LTB patched onto a Robertson-Walker -- the acceleration still has little or no impact, although this depends on the exact model for acceleration, of course.
Something that was in vogue for a good few years was the idea that local structure could account for the observed dark energy, without the need for any negative pressure at all, by whacking Earth somewhere near the middle of a void around a gigaparsec across. This doens't seem particularly plausible anymore, although that's not least because the models studied so far have been staggeringly over-simplified, not because people are resistant (though some are) but becuase the problem rapidly grows impossibly difficult. Literally impossibly.
"do you think that a parsimonious everywhere-the-same negative pressure ... [is] dead for DE already"
I'm certainly not going to state that this is dead. The simplest model is a cosmological constant and there's no reason to state that there isn't a cosmological constant, and some good reasons to say that something that acts as a constant exists. (If nothing else, the low-energy limit of many generalised theories of gravity manifest in the action as basically the first few terms in a Taylor series for the Ricci scalar -- so where general relativity has R, genearlised theories can be written as C + R + (1/2)*c*R^2 + ... where C and c are constants, and those three dots hide a world of unhappiness. Interestingly, C here would act as a cosmological constant, and the R^2 term acts exactly as inflation and, in fact, is both the earliest studied inflationary model (Starobinsky in the late 1970s, a good few years before Guth, albeit with very different motivation) and is slap in the middle of the allowed parameter space. No other simple model is in as good agreement (though it must be pointed out that other simple models are within one sigma, so this isn't, strictly speaking, necessarily at all important.)
The point then is that the low-energy limit of a generic gravity has a cosmological constant, and therefore acts to accelerate the expansion of the universe. More significantly, this is a *screened* constant, meaning that there may well be a fundamental constant in nature, which we can call L, and a constant coming out of an approximate description of a better model of gravity, which we can call, I don't know, A. The observed constant in the action is C = A - L. (That negative sign arises for entirely tedious and unimportant reasons.) So even the naturalness problem, which queries why the observed constant is so small is, if not removed, at least bounced elsewhere.
What's even more, the low-energy limit of a generic gravity also gives us a theory of inflation that fits the data perfectly. *And* it gives us normal gravity. And it does all of this without any scalar fields.
Brilliant, right? Well, half-convincing at least, but there are the usual caveats and problems, not least that this is very definitely a phenomenological description - we don't have the theory this is meant to emerge from. Bummer.
Anyway, this digression was tho point out that there's no way I can say that an everywhere-the-same negative pressure doesn't exist... because I think it *does* exist, in the form of a screened cosmological constant. There may also be a negative pressure that is almost the same everywhere coming from some scalar field, probably an effective field rather than a genuinely physical field.
The presence of these negative pressures then changes the smaller-scale solutions I was talking about -- so you have a LambdaLTB or a phiLTB (Lambda with a cosmological constant, phi with a scalar field), which behaves slightly differently to the usual dust-only LTB. Likewise with Schwarzschild -- Lambda Schwarzschild is normally known as Schwarzschild-de Sitter. And so forth. But even so the influence of a negative pressure on these metrics is very different to its influence on a Robertson-Walker, and the observed acceleration is attributed to a feature of Robertson-Walker, which is only valid on the very large scales.
Basically what I'm saying is that the universe is made up of until billions upon billions of metrics that near to the objects that made them are close to Schwarzschild (or Kerr, or Kerr-Newman), and when you look at these billions upon billions of metrics overlaid on one another they "average out" (in some vague, unspecified and actually appallingly ill-defined manner -- doing this this is an absolutely unsolved problem) to a Robertson-Walker. The fundamental issue in cosmology I'm talking about isn't really whether or not there is negative pressure, since there almost certainly is, but that we're modelling its effects with a Robertson-Walker metric on the unproven (and, fundamentally, unfounded) assumption that not only do the metrics of the matter in the universe "average out" to Robertson-Walker but also that the *dynamics* governed by those metrics is the same as the dynamics of Robertson-Walker. Neither of these statements can possibly be true, although the former seems likely to be at least a very good approximation and the extent to which the latter is wrong is very much open to debate because while I'm certainly right that this is a fundamental issue and that cosmology is in principle totally wrong, it may very well also prove that the error introduced is minimal. And speaking emotionally, I think unfortunately that that's going to be the case, and we're going to be left floudnering around continuing to wonder what the hell is producing such a large negative pressure.