Safari Home Screen + Web Push notifications

Hey, I have been trying to view my website on my iPhone's home page, I added it to home screen, however, every time I press on it, it reopens Safari in the url and does not open the website as if it is an app like how some other websites are. The reason I wanted to do this is because I wanted to test out my website's push notification on mobile apps and according to sources, I have to add it to my home screen before anything. Then it said the second step is allowing Push API within the Safari Advanced tab in Settings, however, for me the Push API does not even show up, I don't know if there is some step I'm supposed to enable or is it because Push API is not featured in iOS 17?