How can I make a mobile app that is basically a mobile browser to access my live Drupal site? There is a question about exporting Drupal pages for use in a mobile app, but I would like to know if there is an easy way to make a mobile app (iOS, Android, etc.) that simply loads the existing site in a browser for online-only access.

I looked into Phonegap, but (as far as I can tell) you have to export the pages as HTML for that.  However, the primary purpose of my site is to send messages back and forth using the Private Message module, so making the site accessible offline makes no sense.

I already have a responsive theme, so the site also looks decent in mobile browsers.  I just want a mobile app because I want the site to appear in App Store and Google Play searches.

Is there a straightforward way to address this use case?
There is no module(convenient) to build all apps in Drupal, You have to build your own or if you have a mobile site then you can do it without Drupal help, see my article for android app. There are so  many builders like iBuildApp, Appypie, Como, and so many.