Can the screen recording be started some other way than from the control center?

To my knowledge that is the only place that you can put the icon. Is there a reason you wish to start it from somewhere else? If you are concerned about Control Center being visible, as soon as you tap the icon to start the recording, it counts down from 3, and that should give you ample time to close Control Center.

Glad to help. I will use it sometimes to provide help to people when using the iPhone, and that makes it easy for me to start on a particular page, then start the recording, but be off the Control Center before it starts. Makes it look a little more professional. Enjoy.