Could ANY cipher serve as the foundation for a public key cryptosystem? My reasoning goes like this:


In principle, any trapdoor function can serve as the foundation for
an asymmetric encryption scheme.
All ciphers are trapdoor functions, since it's easier to encrypt plaintext than it is to crack ciphertext.  Put another way, it's easier to use key and plaintext to produce ciphertext, than it is to go backwards and use ciphertext to produce key and plaintext.
Therefore, one could (at least in theory) devise a public key
cryptosystem derived from any cipher, whether it be Caesar or AES.


This has the interesting implication that one could devise an asymmetric encryption scheme based on an existing asymmetric scheme + key + plaintext combo, and then derive another from that one, etc. etc.

I found this interesting quote which seems to confirm my hunch:


  every problem X in NP has a naturally associated public-key cryptosystem (warning: it's a .PDF file)
Your reasoning is flawed. Producing a cipher text for a given key from plaintext without the key is no easier than the reverse for symmetric ciphers. 
Without the key k, producing ciphertext $Enc_k(x)$ is every bit as hard as $Dec_k(x)$ for many ciphers it is the same operation entirely (e.g anything in CTR mode). And even for common block ciphers there is great similarity between encrypt and decrypt. 

For a public key system I need some operation which can be done without the private key. AES etc. do not supply one. If you have the key both operations are easy, without the key they are darn near impossible.