Look at your schematic.
Level shifter LV input 3 not connected , HV 3 connected
LV input 4 connected HV 4 not connected
How does that work ?
Also, you don't show any current limiting resistors for the leds . Why ? (should be 220 ohms)
Why are there no 4.7k pullup resistors on SDA & SCL ? That's a standard requirement. I2C backpacks and lcds have them built in but when you DIY you have to include them to keep the bus stable and biased correctly.
I forgot to ask, does it matter what clock frequency the ATtiny85 is set for in the boards.txt file ? (will 1 Mhz work ?)