I take your point about primes being closely packed at low numbers, but I think this is a small correction (i.e. you might expect 8-9% of primes ending in 2 to be followed by another prime ending in 2, but certainly not <4%)
https://news.ycombinator.com/threads?id=crnt2
The result - the factor is actually surprisingly large!
I did my own investigation using base 3 and noticed something peculiar.
In the first 100k primes, we go from 1 to 2 29028 times and from 2 to 1 29029 times.
Then I filtered out the twin primes since those are the ones that exploit the fact that the next "possible" prime is one that flips the last digit.
This filtered out 10249 primes going from 2 to 1 (bringing the total below both the number of primes that stay at 2 (21008) and the number of primes that stay at 1 (20932)).
It didn't factor out any primes that go from 1 to 2. Are there no twin primes (p,p') where p%3 == 1 and p'%3 == 2?
edit: Oh hey, this is obvious, if p%3 is 1 then p+2 is divisible by 3. It does mean we don't need to take measurements to know that the result we are investigating cannot possibly account for everything since it isn't a factor at all when going from 1 to 2.
By definition, no there aren't. Twin primes are a distance of 2 apart, so your mod 3 options are 0 -> 2, 1 -> 0, and 2 -> 1. Anything involving 0 means it's divisible by 3, so your only mod 3 possibility for twin primes is 2 -> 1.
EDIT: Excluding the possibility where 3 mod 3 = 0, of course. This allows a 0 -> 2 transition with (3,5).