Actually nowadays it's not that bad anymore. Android browser itself offers installable PWA and there is an event called beforeinstallprompt event (https://developer.mozilla.org/en-US/docs/Web/API/Window/befo...), which can be used to perform PWA installation on user interaction. Of course it's not supported in every browser.
iOS is more difficult since user needs to understand that "saving to home screen" is same as installing "app" and there's no way to trigger it programmatically or help user in any other way than with visual illustrations.
The share menu is used for everything on ios, I had to get used to it at my first apple device’s case.
One notably stupid usage of the Share option was (I believe it is no longer how it’s done) adding an image to a hidden folder — that’s something you definitely don’t want to share, yet quite easy to accidentally send to someone during this process.
Thankfully this insanity is changing in the upcoming release due in September. A button for a 3 dot/hamburger menu appears in the bottom right when you select photos.
Because it doesn't make sense that it's part of the web share functionality to begin with.
The native share dialog is about "I have this piece of content, now share it with one of these other apps", e.g. saving a document locally, or sending it with AirDrop, or saving to Google Drive, or sending as an email attachment, etc.
Installing a site as an app on your homescreen has absolutely nothing to do with sharing content to begin with.
iOS is more difficult since user needs to understand that "saving to home screen" is same as installing "app" and there's no way to trigger it programmatically or help user in any other way than with visual illustrations.