Is it possible to create a button that makes a person add the site on the mobile screen?

Yes. But it depends on the implementation of the service worker browsers ( see here ) and support for web manifestos .

Chrome is the only one supporting both:

• The site must have been visited more than once, more than 5 minutes apart.

• Register a service worker :

  if ('serviceWorker' in navigator) {
    console.log("Will the service worker register?");
    navigator.serviceWorker.register('service-worker.js')
      .then(function(reg){
        console.log("Service worker registrado.");
      });
  }
  

• The site should be HTTPS (because of service worker ).

• You should have a web app manifest as in this example ( the attributes are required):

  {
    "short_name": "AirHorner",
    "name": "Kinlan's AirHorner of Infamy",
    "icons": [
      {
        "src": "launcher-icon-1x.png",
        "type": "image/png",
        "sizes": "48x48"
      },
      {
        "src": "launcher-icon-2x.png",
        "type": "image/png",
        "sizes": "96x96"
      },
      {
        "src": "launcher-icon-4x.png",
        "type": "image/png",
        "sizes": "192x192"
      }
   ],
   "start_url": "index.html?launcher=true"
  }
  

You can add a onclick event to a button:

window.addEventListener("beforeinstallprompt", ev => { 
  ev.preventDefault();
  addHomeScreenBotao.onclick = () => ev.prompt();
});

Other browsers have libraries to simulate this functionality: link