Are string palindrome questions practically interesting?
I had a use for palindromes as follows:

A string of length $n$ and its reversal have the same complexity. Thus, when studying complexity of strings you can identify a string with its reversal.

Now after identifying you may ask how many equivalence classes have certain properties. And then the number of equivalence classes is directly related to the number of palindromes. If there  are $p_n$ palindromes then there are
$$p_n+(2^n-p_n)/2$$
equivalence classes of binary strings.