fm:
I can confirm that the pcf8475 is not able to drive the LCD backlight. It may be possible with a big resistor in between, but it is useless.
It is dependent on the LCD and how bright you drive the backlight.
I've seen a great variation of backlight supply current in lcd datasheets,
particularly between 16x2 and 20x4 displays.
The pcf8574 output pin is rated for 100mw which is 20ma at 5v so if the LCD backlight can light up
bright enough with less current than that, it can work.
I have a 16x2 LCD with a built in resistor that only uses a bit less than 3ma to light the backlight
with no external resistor, so it would work for that LCD.
When I was modding the I2C LCD extra i/o board for backlight control,
I actually did consider just wiring the pcf8574directly to the backlight anode,
But since the 20x4 data sheets I've seen show using considerably more current than
that (generally above 50ma and as high as a couple hundred) to light their backlight,
I opted for using the mosfet to drive the BL pin and go through the 121 ohm resistor
to the anode to try to ensure it would work with any LCD I might use in the future.
--- bill