Consider the set $S_x$ of the following Pochhammer numbers:
$$(x)_n := \frac{\Gamma(x+n)}{\Gamma(x)}\,, \tag{1}$$
with the gamma function:
$$\Gamma(n) := (n-1)!\,. \tag{2}$$
From "experiment" I noticed that the distribution of digits in the sets defined above satisfies Benford's law (see in particular the generalization to digits beyond the first). Consider e.g.:
$$S_1 = \left\lbrace 1,2,6,24,120,720,5040, \ldots \right\rbrace\,. \tag{3}$$
Is it possible to prove non-experimentally that such a set satisfies Benford's law? I am okay with using scaling invariance if it helps.
By definition, Benford's law is honoured when the fractional part of a positive random variable's logarithm is uniformly distributed. This is one of those "often approximately true" results, like a large sample's mean being almost Normal.
You mentioned scaling invariance. If we measured lengths over orders of magnitude, our Bayesian prior for the fractional part will be the same whether we work in feet or yards, and switching between these scales the raw value and shifts its logarithm. The fractional part "wraps around", implying uniformity (see also here).
When $n=1$, your experiment looks at factorials; when $n=2$, it looks at them again, but with the sequence shifted; with any larger fixed value of $n$, the sequence is also rescaled. So we expect the distribution not to depend on $n$, which forces it to (approximately) follow Benford's law.