Secret key and Decryption oracle in the revised Fujisaki-Okamoto transformation
As you say, CCA proofs are actually reductions to underlying problems. In all CCA proofs that I can think of at the moment, the underlying problem is a weaker security notion for an "embedded" encryption scheme - e.g. Cramer-Shoup and friends use IND-CPA of ElGamal and Fujisaki-Okamoto uses OWE of the contained scheme.

The general proof strategy is to take a public key from your challenger and pass this to the adversary. This lets you use the challenger to process the two challenge messages to get the challenge ciphertext - since the only bit of information that your adversary gives you is which of the two challenge messages he thinks was encrypted, I'd guess that all CCA proofs will somehow have to "embed" the hard problem they're reducing to in the challenge ciphertext. After all, the reduction could compute everything else by itself.

This strategy of getting the public key from the challenger means you don't know the matching secret key, so the interesting bit of the proof is then how you handle decryption queries without it. This is certainly general to all CCA proofs that I can think of at the moment.