Is there an iPhone SE simulator for Xcode 11, iOS 13?

Also can be useful. The problem may be in your project preferences. If downloading simulators didn't work, try this. Open you project in Xcode -> Choose your project on the Navigator which is on the left side Navigator

Then in the Deployment Info in target tab choose IOS version for your project (choose 10 or 11 for all simulator displaying)

Deployment Info

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.

Edit: This is still valid with XCode 12.3 and iOS 14.3 as of January 2020