Ah-ha. Using the I2C interface for high level commands is fine. They get sent via interrupts anyway. So you can get on with your own application.
Yes, you can have a secondary AVR that receives commands via I2C and does the donkey work for the display. As far as I know, Digole use a PIC cpu for this purpose on their intelligent monochrome displays. And can use UART, SPI or I2C for receiving the commands.
Several makes produce intelligent full colour TFT displays.
The PCF8574 backpacks are not intelligent but they are cheap.
I will have a look at the message link from the OP when I am at a PC. Clickable links can be used by Tablet or PC.