In what practical sense are websockets overkill? As far as I know, the resource usage isn't really all that different. It also isn't as if websockets are that difficult to use these days.

I am also skeptical of the "doesn't need websocket" statement. It is technically true that you can use other technologies. But, WebSockets provide bidirectional communication, while Ajax Polling and Long Polling are used for periodic updates. HTTP/2 Push is more suited for pushing resources or data from the server to the client. As far as my understanding goes the use case there is specifically aimed at pushing resources the server "knows" the client will need as well without the client requesting them. I am sure that with some creative code you can also use this to implement a chat of sorts but that is a bit besides the point.