On chrome on mobile devices, the install button doesn't appear every time. This depends on machine learning algorithm as explain here :
developer.chrome.com/blog/how_chrome_helps_users_install_the_apps_they_value

To help user to install the PWA without using "Add to home screen", you can try to use the event beforeinstallprompt.

There are some information here : https://developer.mozilla.org/en-US/docs/Web/Progressive_web_apps/Guides/Making_PWAs_installable#triggering_the_install_prompt

And next is an example of code from https://whatpwacando.today/installation

let deferredEvent;

window.addEventListener('beforeinstallprompt', (e) => {
  // prevent the browser from displaying the default install dialog
  e.preventDefault();
  
  // Stash the event so it can be triggered later when the user clicks the button
  deferredEvent = e;
});

installButton.addEventListener('click', () => {
  // if the deferredEvent exists, call its prompt method to display the install dialog
  if(deferredEvent) {
    deferredEvent.prompt();
  }
});

These days you can

use Bubblewrap to package your Progressive Web App for Google's Play store

More info here: https://developers.google.com/codelabs/pwa-in-play#0

And here: https://developer.chrome.com/docs/android/trusted-web-activity/quick-start/