the current drain will still be very similar,

Let's think it through. 5ma per segment and you have 2 segments -> 10ma per digit when that digit is on, 0ma per digit when that digit is off.

Let's say that you "mux" them 1ms per digit.

ms 1: digit 0 is on, digit 1..7 are off. Total current consumption: 10ma (for digit 0).

ms 2: digit 1 is on, digit 0, 2..7 are off. Total current consumption: 10ma (for digit 1).

ms 3: digit 2 is on, digit 0..1, 3..7 are off. Total current consumption: 10ma (for digit 2)

...

Do you see a pattern now? What's the new math?

I guess the remaining opinion I hold is that it is possible that there is a driver on board the LCD

You guessed right. Many of those lcds have backlight rated at 20ma max and no big difference at 10ma.