Can I "use" my iPhone SE, through my MacBook Air?

This isn't a complete answer because there's no simple way to really control the phone via mouse and keyboard. I can get you past that screen though.

First, plug your phone into your computer.

Then open QuickTime Player and choose "New Movie Recording". Using the little drop down right next to the record button, switch it from the built in camera to the name of your phone. You don't have to actually record it, but this will allow you to see the full screen.

Next, go to iTunes and open the page for the device (this button: ). Then scroll down in that Options part and click "Configure Accessibility…". Under Seeing, choose VoiceOver, and then click OK.

Now you can use your phone, but it's going to be tedious to type.

To cycle through all the buttons on screen, swipe left and right. To click one, double tap on the screen. This includes the keyboard keys, so you can slowly type the wifi password. (However, you can hold off on typing that in, and skip this screen for now. You will be able to use the Mac's keyboard to type)

If you download Mocha Keyboard Lite to your computer, and on your phone, use VoiceOver to go through the Bluetooth settings, you can enable Bluetooth and connect to your computer. That way you can use your MacBook Air's keyboard to type on the phone. That's as close to normal usage of the phone as you can get with a half-broken screen.

^{Note that the lite version of Mocha Keyboard does not allow you to use the delete or backspace key! Type carefully!}

^{Edit: if you really need to use backspace with Mocha Keyboard Lite, just press Control+Shift+B, then Control+K. Don't ask me how I know this!}