Let's say I want to use ECs to do asymmetric encryption. Suppose both sides have generated EC keypairs and have exchanged their public keys. I run ECDHE, and get a derived secret. I would then either use that as a key for AES, or create a derived key with an incrementing nonce for subsequent "sessions".
My gut tells me rolling my own is a bad idea. Not having seen this construct implemented in any crypto library tells me it's an even worse idea. Is there a concrete flaw in what I've described above?