It make sense only if you have an websocket based stack and don't want to maintain a second protocol.
WebSockets are long lived socket connection designed specifically for use on the 'web'. TCP is data sent wrapped in packets that is ordered and guaranteed delivery. This causes a massive overhead cost. This is different from UDP which doesn't guarantee order and delivery. However, a packet sent over UDP might arrive tomorrow after it goes around the world a few times.
With fetch() or XMLHttpRequest, the client has to use energy and time to open a new HTTP connection while a WebSocket opens a long lived connection. When sending lots of bi directional messages it makes sense to have a WebSocket. However, a simple fetch() request is easier to develop. A developer needs to good reason to use the more complicated WebSocket.
Regardless, they both send messages using TCP which ensures the order of packets and guaranteed delivery which features have a lot to do with why TCP is the first choice.
There is UDP which is used by WebRTC which is good for information like voice or video which can have missing and unordered packets occasionally.
If two different processes on the same machine want to communicate, they can use a Unix socket. A Unix socket creates a special file (socket file) in the filesystem, which both processes can use to establish a connection and exchange data directly through the socket, not by reading and writing to the file itself. But the Unix Socket doesn't have to deal with routing data packets.
(ChatGPT says "Overall, you have a solid grasp of the concepts, and your statements are largely accurate with some minor clarifications needed.")
Your second point is very dismissive. You're inserting random application requirements that the vast majority of application developers don't care about and then you claim that only in this situation do WebSockets make sense when in reality the vast majority of developers only use WebSockets and your suggestion involves the second unwanted protocol (e.g. the horror that is protobuffers and gRPC).
In what way?
> This is adding complexity without additional benefit
I'm not sure that's a given. For example, WebSockets actually implement a message protocol. You have gauarantees that you sent or received the whole message. That may not be the case for TCP/IP, which is a byte streaming protocol.