I should also add: I don't install apps like Signal/Whatsapp on the SIMless phone. No one should be able to call me on it. Period.

I do have a mail app on it, and it checks mail only when I tell it to (i.e. not running in the background giving me email notifications).

At home, I keep the phone with the SIM in one room. I can use the SIMless phone around the house and not worry about pings.