So if you want to share your MMPK with the public, then you need the Publisher extension. If you have the Publisher extension, then when you are in Pro you will be provided with the option to "Enable Anonymous Use", which you must take. If you don't see that option then you don't have the extension. If that is the case then users will need to sign into the app with a named user account.
Another reason your MMPK may not be showing is because your TPK is in a different projection to that of your feature layers and map. It is best to use the WGS 1984 Web Mercator projection for map and TPK to avoid any incompatibility issues. You can use other projections if need be but the map and TPK need to be in same projection, but I have occasionally seen problems arise with custom projections.
The workflow you described is normal in that people first need to download the map before going into an area without coverage. But once it is offline, they wont have to do so again.
Do check out the TPK projection and Publisher extension, but if that is not the cause, do check out the Map Viewer template. As of AppStudio Version 5, you can now also edit features in this app.