No, because CORS can only restrict which origins (scheme, domain, port combinations) are able to access the site's data. But you're not even connecting from a web origin but from localhost and you're trying to defend from all access except by your frontends. For this, you need a shared secret between the server and the frontend.
A further limitation of CORS is that certain requests are allowed even if they are not from an allowed origin.
I assume someone could have a look at the JavaScript on the browser and see hey this must be the secret stored here because it is passed to the server on every request. Then write there XSS attack to use that.
One problem is that we developers are lazy and use Access-Control-Allow-Origin: * (wild star) instead of actual hostname. (eg. allowing all origins to access the backend)
No modern browser allows access to localhost without that header.
But it's still possible to forge a request using curl or whatever to bypass CORS.
So as the parent post suggest - use a token of some sort.
I also recommend using a strict Content-Security-Policy to stop X-site injection attacks. (eg someone adding an image to your page/app with src="/api/cmd=rm -rf /"
A further limitation of CORS is that certain requests are allowed even if they are not from an allowed origin.
To conclude, you definitely need a secret.