I'm running Xcode 11.0 on macOS Mojave (10.14.6) and though the iPhone SE is officially supported on iOS 13, it doesn't appear in the list of simulators.
It can't be added nor downloaded, as far as I can see. Any ideas?
Xcode 11 doesn't automatically add every possible simulator. But you can add an iPhone SE iOS 13 simulator.
In Xcode, click on the Window menu and select Devices and Simulators. Select the Simulators tab. Tap the + in the bottom-left corner. Give the new simulator a name. Select iPhone SE as the device type. Enter iOS 13 as the OS Version.
Now you have what you need.