I seriously doubt that most laypeople think that.

Why?

You can get texts, app notifications, and phone calls, you can play music and podcasts, you can download app updates, an app you opened earlier is still in the same state later. All with the phone screen off. They also know that none of these things are possible when their phone runs out of battery and dies. I think most people could notice the difference between these states and realize that screen off != phone off.

What I think most people don't notice is that the fact that simply because they can receive phone calls and notifications implies that they are in practice being tracked.

I've done end-user support for laypeople for 30 years - including their mobile devices. The least tech-savvy of them understands the phone is still on, after the screen goes dark.

Most laypeople would understand the tinfoil part of a tinfoil hat.

I have a faraday bag I put my phone in when I am traveling and don’t need it. It blocks all RF. It’s meant for forensics and is of very high quality.

I found that "if someone calls you, will it ring?" is a question that works to demonstrate that it's not actually off.