I may be mistaken, but I do not think that is possible. I tried creating a simple Gmail add-on with a simple "splash" card, and then used Utilities.sleep(3000) [3 seconds] to call another function. I could not get it to work at all, it just gives an error stating "Unable to display the content".
I think this is limited because all widgets, events, and actions are very tightly controlled and require user input. Otherwise we could just execute javascript any way we wish including deleting user emails without their knowledge, deleting, copying or corrupting drive files without user knowledge, etc etc.
If you want a Splash screen, I suggest adding a simple button on it that says "Start" or something. As far as I can tell, all "events" are user-initiated, and all triggers are defined in the Manifest. If you can figure out how to set a timer function or something, I'm guessing it would be via the manifest.