The crucial element here is that this indeed needn't be
> more ways for a site to shove things in front of me
The prompt can be provided by the browser, at a moment of choice for that browser, in a standardised way, and with the option to be disabled globally. For example, a browser could choose to only show the option once for websites you use often and provide a manifest file, and/or when you bookmark them, and/or just show an icon in the address bar. So many ways to do this in a nonobtrusive way, that still makes sure that people who would be interested in adding it to the home screen (i.e. a subset of the people installing an app today) actually know they can do that.
> more ways for a site to shove things in front of me
The prompt can be provided by the browser, at a moment of choice for that browser, in a standardised way, and with the option to be disabled globally. For example, a browser could choose to only show the option once for websites you use often and provide a manifest file, and/or when you bookmark them, and/or just show an icon in the address bar. So many ways to do this in a nonobtrusive way, that still makes sure that people who would be interested in adding it to the home screen (i.e. a subset of the people installing an app today) actually know they can do that.