ECC can be used to encrypt the data itself with the public key instead of encrypting a random key and then symmetrically encrypting the data with that random key. But this is not a good idea as explained in Thomas Pornin's answer to In PGP, why not just encrypt message with recipient's public key? Why the meta-encryption?.
In any case ECIES uses a symmetric encryption algorithm to actually encrypt the data, so even if you use ECC to encrypt all the data, you're still encrypting it with a random symmetric key. But if you're encrypting the data itself with ECIES you could use a simple XOR as the symmetric encryption algorithm.