2 added 218 characters in body
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Brachylog, 14 bytes

1|r?$ph:?r/:0&

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Explanation

This implements the formula explained in the challenge description.

Computing all products of suffixes of the prime factorization and checking that they are all palindromes is 1 byte longer (1|$p:@]f:{*.r}a).

1                  Input = 1
 |                 OR
  r?               Reversing the Input results in the Input
    $p             Get the prime factors of the Input
      h            Take the first one (the biggest)
       :?r/        Divide the Input by that prime factor
           :0&     Call this predicate recursively with that new number as input

Brachylog, 14 bytes

1|r?$ph:?r/:0&

Try it online!

Explanation

1                  Input = 1
 |                 OR
  r?               Reversing the Input results in the Input
    $p             Get the prime factors of the Input
      h            Take the first one (the biggest)
       :?r/        Divide the Input by that prime factor
           :0&     Call this predicate recursively with that new number as input

Brachylog, 14 bytes

1|r?$ph:?r/:0&

Try it online!

Explanation

This implements the formula explained in the challenge description.

Computing all products of suffixes of the prime factorization and checking that they are all palindromes is 1 byte longer (1|$p:@]f:{*.r}a).

1                  Input = 1
 |                 OR
  r?               Reversing the Input results in the Input
    $p             Get the prime factors of the Input
      h            Take the first one (the biggest)
       :?r/        Divide the Input by that prime factor
           :0&     Call this predicate recursively with that new number as input
1
source | link

Brachylog, 14 bytes

1|r?$ph:?r/:0&

Try it online!

Explanation

1                  Input = 1
 |                 OR
  r?               Reversing the Input results in the Input
    $p             Get the prime factors of the Input
      h            Take the first one (the biggest)
       :?r/        Divide the Input by that prime factor
           :0&     Call this predicate recursively with that new number as input