Manifest is not managed by iOS yet (in development currently). There is an interesting library to add the essential elements for iOS automatically. But there isn't no way to show a home screen pop-up at this moment...
You can follow the status about features on iOS here : https://webkit.org/status/#specification-web-app-manifest
Could you refer following link https://www.monterail.com/blog/pwa-for-apple-ios?
In that link, he already mentioned that
Once you enter a website, you won’t be greeted with the Web App Banner and you will need to use “Add to Home Screen” option in the share sheet. As you can in the screenshots below, there’s no sign that you’re adding a PWA rather than a website.
Now what you need to do it, you need to check current user-agent is
safari device, you better display popup dialog that saying Click share button and click Add to HomeScreen button to install your app at device.