When you Run Locally, does it run the native app successfully? At first a message is displayed when web part is ready. Then second success message popup appears for the native app which says that the app is available to be viewed on mobile (what we do with make it native app)
In your project directory, see packager log text file for any errors in it
A similar kind of issue I have experienced was solved by updating the Windows. At that time it required Windows 10 build 1909. Please check for windows update and install latest ones. Restart your computer and then run the app.
Check to see if NodeJS has proper rights on your local machine through your firewall and if Watchmen.exe is running after deployment (it should).