  Using quantization techniques, we show that the $\delta$-invariant of
Fujita-Odaka coincides with the optimal exponent in certain Moser-Trudinger
type inequality. Consequently we obtain a uniform Yau-Tian-Donaldson theorem
for the existence of twisted K\"ahler-Einstein metrics with arbitrary
polarizations. Our approach mainly uses pluripotential theory, which does not
involve Cheeger-Colding-Tian theory or the non-Archimedean language. A new
computable criterion for the existence of constant scalar curvature K\"ahler
metrics is also given.
