The $/ =~ y///c bit, then, counts the number of characters in $/, the input-record-separator, by replacing everything, all chars in the input-record-separator (here described as the complement of nothing) with nothing and returning the number of chars thus replaced.

Not quite; since the REPLACEMENTLIST defaults to the (post-complementing) SEARCHLIST (except with /d), all the chars are replaced with themselves, not nothing. So $/ is unchanged. (Actually, tr aka y recognizes when it's only being used to count and can even be used on readonly strings then.)