You can encrypt with secp256k1 (or any elliptic curve) by using ECIES (Elliptc Curve Integrated Encryption Scheme). However, inside ECIES a symmetric key is generated and the message is encrypted using it.
This is in contrast with RSA, which can encrypt data directly; though most of the time it is also used to encrypt a symmetric key and then encrypt data with that key.
If you really want to avoid AES, then you can use the secret derived inside ECIES and simply XOR with your message. Check the SEC 1 standard, section 5.1.3.