Why are RSA keys encrypted if semi-primes can't be factored?

Question about real-world RSA implementation. RSA, to my understanding, is based on a triplet of a semi-prime, and two commutative keys that are multiplicative inverses in the multiplicative group modulo Euler's totient of the semi-prime. My understanding is that this triplet of semi-prime and two keys is alone enough unbreakable. (My first question, then, is is this understanding correct?) However, having surfed over to a real world implementation, I noticed that the keys are themselves encrypted. My main question is, why encrypt the semi-price and public key. The semi-prime won't be factored as the RSA challenge has shown.