It looks like the Online Encyclopedia of Integer Sequences has some information on your series [1]. Always an interesting place to start going down a rabbit hole.
Fascinating stuff... The palindrome observation of the OP might just be that we are good at finding patterns and there are limited numbers of distances with lower numbered primes (and from the graph of your reference it looks like a lot of the distances are concentrated in smaller numbers -- max dist increasing logarithmically?)
I wonder if the palindrome conjecture can be posed mathematically or algorithmically (seems like it should). It would be interesting to test palindromes to a certain number compared to palindromes of a similarly distributed set of random numbers. Counting all palindromes of any size will probably be a significant big O.
Now I'm just trying to keep myself from looking at the Collatz Conjecture. :)
Prime series are palindromic. Given the set of primes (2, 3, 5, ..., p[m]), there is a repeating pattern of length r = 2 * 3 * ... * p[m], where nr +/- [ p[m+1], p[m+2], ... p[m+q] ] (p[m+q] < r/2) are relatively prime to the given set. When n = 1, the only relatively prime integers which aren't truly prime are those which are some combination of two or more p[m+1] ... p[m+q]; the smallest is of course p[m+1] ^ 2.
[1] https://oeis.org/A047160