In the worst case scenario with server-push that you described, the server decides to push content the client already has cached before pushing content the client doesn't have, and begins transmitting that redundant content before the client has a chance to cancel the push. Once the client receives push promises for the content it doesn't need, it can immediately send a request to cancel the download of those resources. Once those cancellation requests reach the server, it immediately stops transmitting the resources the client has cached, and starts transmitting the data the client actually needs.

Contrast this with the same scenario without server push, where the client parses the HTML for the main page to determine what resources it needs, then send requests for those items which the server must receive before it starts transmitting them. In both cases a round-trip from the server to the client and back is needed before the server can start transmitting the necessary assets, but without server push the client needs to download and parse the main page before it can start telling the server which resources it needs, whereas with server push it can send cancellation notices before the current page is downloaded or parsed.