You could indeed do that, it's just more work. The beauty of server-side rendering a React app in a headless browser is that since the environment is the same as what the app was originally designed for (a browser), things _just work_.
For example: you load the app in your headless browser and let it load as would in any browser. The browser fires some events and XHR/WebSocket requests and you can cleanly wait for them to finish as opposed to doing something like this: https://techbrunch.gousto.co.uk/2016/10/10/isomorphic-react-... which requires you to do some extra configuration in your app to help support a Node environment.
TL;DR: ReactDOMServer.renderToString won't wait for your async requests to finish, you need them to execute and finish before you call renderToString