Why can't Cross-Origin request not share cookies instead of denying the request?

I'm working on a simple SPA website that consumes a REST API from another website. Because this site doesn't have a policy to allow CORS requests, I need to use a proxy, however, the company I work for blocks most of them, the ones I found to work have a limit on the number of requests which is a bummer, and I didn't want to host my own as this should've just a simple frontend app.

I understand why CORS policies are needed, you can't simply share cookies with everyone that sends a request from your browser to a given host.

However, instead of completely blocking the request, why can't browsers simply don't share such cookies and other info in those cases where CORS applies? (which is the result I get if I send this request from a backend).