Answer
What are the criteria for "Add to Home Screen" to appear?
Chromium browsers offer "Add to Home Screen" once a site is served over HTTPS, links a manifest with at least a name, a 192px and 512px icon, a start_url and a standalone/fullscreen/minimal-ui display mode, and has registered a service worker with a fetch event handler. Some engagement (visiting the site more than once, with time between visits) is also typically required before the automatic prompt fires.
The technical checklist
HTTPS, valid manifest fields, registered service worker — verifiable instantly in DevTools → Application → Manifest, which lists any missing criterion.
The engagement heuristic
Chrome does not prompt on the very first visit; it wants signs of a returning, engaged user before offering install, to avoid prompt fatigue.
iOS has no automatic prompt
iOS never fires an automatic install prompt — users must always use the manual Share → Add to Home Screen flow regardless of criteria.