You have to consider both data at rest and data in transit here.
What about data in transit?
Data in transit is data that is being exchanged over the network between the client and the server. For the purpose of security, there is no difference between potentially sensitive data being sent as a cookie or in the request/response body. All of it is part of the message, so you have no advantage whatsoever between sending the content as a cookie or as the message body.
What about data at rest?
This is a more interesting part. Cookies are designed to give some sort of persistent storage while browsing the web, which was initially designed to be 100% stateless.
When a cookie is set by a website, the client saves the name, the content, the expiration date and some other metadata about it.
Another side-effect of storing data as cookies is that they remain persistent on the client, even when not in use. This means that an attacker, who is able to read out the cookies, can see the content of the messages.
Is there any advantage of using cookies instead of message bodies?
This question is like asking "Is there any advantage of using a frozen banana instead of a hammer?". Yes, you might be able to drive a nail in, but you also might break the banana. And your hands will get cold.
- It requires extensive overhead, both in freezing the banana and in transforming the "message body" from the cookie back to a regular message body.
- Such systems have not been tested as much. I trust a hammer not to break when hammering a nail into a wooden board, and I trust my browser not to break when receiving a message body. Usually.
- You circumvent a lot of security measures designed for normal systems. For instance, for the purposes of your browser, all content is added "dynamically" to your DOM, which means it's treaten differently than content that was received in the message body.
In general, I would not do it. If there was an inherent flaw in sending data via message bodies, the system would be updated. This idea probably stemmed from someone who believes that anything that is considered a de facto standard is inherently bad and evil, and only TheirOwnSolution™ is the only secure way. It's generally safe to disregard their opinions.