An update on this: it turns out there are two different problems with the launcher app as a result of security changes introduced in macOS 10.15 Catalina.

The first is that the launcher did not execute files launched with it (for example, by double-clicking on the source file or by dragging the file to the launcher icon).  The launcher launches but no Terminal window opens.  That problem has been fixed in the just released python.org macOS installer for 3.8.3 and will be fixed in the next releases of 3.7.x and 3.9.x (pre-release).

The second problem is that the launcher app is not able to launch Terminal.app to run the script if Terminal.app was not already running. Unfortunately, I was not able to provide a fix for that problem in time for 3.8.3. However, there is a relatively easy workaround until that fix is available. Before using the launcher app, just ensure that Terminal.app is already launched; the launcher will then be able to run any number of scripts in new terminal windows as expected.  One easy standard way to launch Terminal.app is to invoke Spotlight by clicking on the "magnifying glass" icon near the upper right end of the macOS menu bar and then typing the letters "terminal" followed by the Return key in the resulting search window.  When a script is done running, just close the terminal window but don't quit Terminal.app if you want to keep using the launcher.