Use bitcoin blockchain as a way to send messages to another address purposely setting a 0 fee or a very low fee? Having that bitcoin nodes are all over the world. Is it possible to broadcast a transaction sending to 20 different addresses with a piece of the message embedded into each address? 

The goal is to make use of the blockchain unconfirmed transaction mempool to relay this message to the recipient.

The fee would be set to 1 satoshi per byte. We know it will never confirm, but thats not the goal of it.

What are some drawbacks to this?
This is possible, but there is always a chance your tx will confirm.

As a drawback, the blockchain is public, so the entire world will be able to view your message, assuming they know what to look for. You can encrypt and fragment your message, but using the blockchain to transmit it seems inefficient compared to other available options.

What is the advantage to using the blockchain, as opposed to another channel of communication?