When I audit the PWA, I get the warning: Does not register a service worker that controls page and start_url

You get the warning due to restrictions placed by Shopify on service worker scopes. The service worker is a critical part of PWA technology and acts as middleware that processes all network traffic within its scope. It has control over traffic to the site and how it responds. Shopify is examining the implications of having service workers with root scope (your home page) as they control all network requests within your site. In early 2020, they put a restriction on service workers with root scope and we have to work around this restriction which leads to the warning. This is not ideal, but we remain hopeful that Shopify will relax this in time. They already allow it for headless commerce, including through their headless solution Hydrogen.