Why do baseband processors remain on, and communicating with the tower, when the phone is shut down?
that if you really want to turn off a cell phone, you need to take the
battery out

No, that's not true, all the phones I know can be fully switched off, usually by holding down the power button. You're confusing switching the phone off with the standby mode of the phone where it looks like it is off (display off, no lights flashing) but it can still receive a call. In this state the battery should not be drained that much so the phone can stay for months or more in this state.
For receiving a call the &quot;cellular communication part&quot; (that includes the baseband processor) of the phone needs to be activated on a regular basis to check if someone is trying to call you. In this state the battery will be drained such that the phone can stay in this state for a few days or more (simple &quot;dumb phones&quot;can last for weeks) in this state.
All phones are shipped in the &quot;power off&quot; state and that makes sense as otherwise the customer would receive a phone with a drained battery. That's bad for the battery! Also: a new phone does not contain a SIM card so it cannot even fully connect to a network so there's no point to have the baseband processor activated.
TLDR: you're confusing &quot;power off&quot; with &quot;standby&quot; which are different things. So the batteries do not need to be removed. Switching the power off (you might need to hold a button or use a menu) will stop all communication.