The original point of the OP's question was a ELI5 overview of LBM. But, I will provide some relevant works off the top of my head since I am part of a research group on LBE-DNS and LBE-LES. Lallemand and Luo (2000) proved the lower numerical dissipations and small numerical dispersive effects (compared to N-S) , key in small scale details of turbulence. DHIT was being done in LBM in the early 90's (Chen 1992). Martinez et al. (1994) found results with accuracy inline with spectral methods for turbulent flow. Valino (2010) found the same result with 3D. The claims of superior rotational symmetry (better than FVM and FDM continuum) can be found by following the Chapman-Enskog Expansion, a good place is: "Lattice Boltzmann Method and its Applications in Engineering" by Guo and Shu (2013). Yu and Girimaji (2005) found that LBE-DNS is excellent for capture of anisotropy. Even with coarse grids (relative to kolmogorov) the Entropic LBE was able to achieve stability and surprisingly accurate results. The logic behind my statement about less time is the fact that LBE is local and simulates an ODE. When you extend that to high Re DNS it is undeniable that the computation is quicker, all else being equal. You've presented a nonsequiter argument by showing that others have not used LBE. Correlation, or happenstance, DNE causation, and considering NS is 30+ years older, the LBM literature is catching up fast in filling the gap. Lack of use does not determine lack of legitimacy, or even superiority. The spectral methods are fine as a gold standard, but seeing as they rely on a "neat" decomposition of the NS equations and are thus not suitable for arbitrary geometry, my commentary on LBE-DNS superiority was based on comparison with NS FVM,FDM,FEM.

I honestly don't know what you are on about with the stability statement. The statement "improperly specified lattice boltzmann techniques" is meaningless. Anything improperly specified will cause error. The LBE that exists conserves everything. I never mentioned lattice Mach number, and I'm not sure what "trick" you are referring to other than that higher lattice velocity results in a more stable solution. Which certainly is no trick. I really highly suggest you look into MRT, as it appears you are unfamiliar. While relaxation time scales with grid and Re in MRT, Hua and Chang (2006) performed Re 1,000,000 on a 513x513 lattice using MRT and reported being no where close to a stability limit. I'm really not sure how to address much of this issue for you because much of it is not relevant or related to the lattice Boltzmann method, and does not follow from what the LBM is.

What I am stating about superior symmetry is superior spatial rotational symmetry (Guo and Shu (2013)) which is a result of the chapman enskog expansion to prove the LBE results in the NS equations. This is a big facet of LBM because its predecessor, LGCA did not obey gallilean invariance nor spatial rotational symmetry.

In summary, the key factors here seem to be: MRT provides all the stability you need and the statement that LBM is not stable at high Re is absolutely false, the LBE contains superior rotational symmetry and superior dispersive and dissipation properties which lend to the simulation of turbulence, and the superior locality, linearity, and scalability of LBM provides for better computational speeds, all else being equal.