As you already know, it is possible to connect an iPhone or iPad to the Mac and record your screen using QuickTime. Controlling it, however, is already another story.
Now an app has emerged that allows you to do just that, and by Chrome, that is, multiplatform (compatible with OS X, Windows and Linux). But what? It for Android devices.
Your name Vysor and it can be downloaded directly and free from the Chrome Web Store, still in beta. The thing seems to work well and already accepts control by both the mouse and the keyboard.
At least for now, Android needs to be connected to the Mac / PC via USB.