Well, so you write down an initial values surface with that approach and let the matter evolve: either you get something physically plausible or you don't. We have some guides as to what is physically plausible, and of course evidence that any of them is wrong would be very interesting.

More theoretically, the LHS doesn't care about how you arrange the RHS at all. More finely, the metric does not care how you compose the metric. If you do it perturbatively, the leading term could be Minkowski or Gödel for all it cares, as long as the sub-leading terms then present a useful background for small perturbations that remain small (or better, vanish) rather than explode into large perturbations. Inside the solar system, Schwarzschild gets that right, and nobody has shown you need to correct for the global behaviour of the cosmic expansion.

> ... if we move the cosmological constant factor to the right handside ..

Before the 1920s (!) there was a discussion about where to put an everywhere isotropic tension in the EFEs. https://arxiv.org/abs/1211.6338

In section 3, the author quotes Einstein's response about how to determine the density. The part after, "Later, he finishes with..." seems especially germane, and cf. section 4 especially its last sentence (wherein "dark energy" in that context means a source field with a potential and dynamics and a coupling to the other source fields and a set of initial values).

[Heh, typo in the second sentence of the last paragraph of §3, "vale" instead of "value".]

> (matter forces) contribute to the stress-energy tensor

Ok, but if you hold that the CC acts isotropically everywhere you can think about decomposing the curvature tensor into its Ricci (and Weyl) components. Weyl tensor encodes the stretch-squash astigmatism: a spherical volume in the presence of a nonvanishing Weyl start to look more like a U.S. football. Ricci scalar encodes the isotropic change in volume. For a volume, nonvanishing Weyl with a vanishing Ricci is volume-preserving; a vanishing Weyl and a nonvanishing Ricci is a contraction or expansion.

The cosmological expansion at cosmological scales clearly is Ricci scalar rather than Weyl, since matter in the large remains isotropic and homogeneous.

Shuffling \Lambda over to the source side does not change that observation: we don't see broken cosmological-scale isotropy, so we must still have a vanishing Weyl and a nonvanishing Ricci scalar on the LHS. How does that happen? Presumably in the metric, since it's the obvious choice of dynamic field (although of course you could insist on making the RHS as complicated as you like, and there have been cases where having the metric maximally static with gravitational dynamics shuffled into "fake" matter is useful -- the original Hawking Radiation paper did just that, for instance; and of course you might want a quintessence-style field instead of a CC, and bite the bullet on recovering observations through that field's behaviour).

Ultimately what we care about is that we have the correct EOMs, and for test particles the EOMs in our labs and in our experimental platforms elsewhere in the solar system, and observed for astrophysical objects throughout our galaxy and the cluster it's in, are nowhere near the EOMs of FLRW. They are much nearer to Schwarzschild.