State tax filing

"Note that as far as I can tell the TurboTax software completely ignores the value in box 16.Note that as far as I can tell the TurboTax software completely ignores the value in box 16."

It's not just TurboTax, but it's the State of California that ignores the amount in box 16.

Take a good look at California form 540. The amount from box 16 on your W-2 appears in line 12 of the 540. 

HOWEVER, this line 12 amount is not used in any other place on the return. Instead, the 540 starts with federal income and adds and subtracts things to get to the state tax. 

When you entered your W-2 on the federal interview, in the next screen - "Do any of these uncommon situations apply to this W-2?" - there is a check box "Paid Family Leave  - I earned all or part of this income as paid family leave"

This is the first thing that tells TurboTax that you have PFL.

Then, in the California interview, there is a screen that asks you to enter the amount on that W-2 that was actual PFL, i.e., payments from the insurance company.

The amount you enter here is the amount that is removed from CA state income. This is how you get TurboTax to recognize CA PFL so that it is not taxed on your CA return.