I’m about to start a similar project so wanted to investigate. I created a simple demo react-router app (render-create-react-app-demo/App.js at master · johnbeynon/render-create-react-app-demo · GitHub). When deployed the app loaded and I was able to click the links rendered. Visiting a path directly failed (as expected) so I added the route as in the linked documentation and it then loaded successfully via the URL.
Whilst this doesn’t help you, it does at least confirm that the documentation is correct. You should at least see your app and be able to navigate it successfully WITHOUT the rule, it’s only if you enter a URL directly in the address bar does the rewrite rule take effect.