The Hayato ASIC (Atari Lynx II)
The Hayato chip is one of two ASICs on the Atari Lynx II.
The console will not operate at all without both chips present.
The Hayato is clearly responsible for audio, driving the LCD, power on control, and likely reading and running the game code, based on the schematic.
We know from this discovery that although the Hayato drives the LCD, the pixel data it puts out on the DLx pins actually comes from the Suzy.
Confirming Hayato Crystal
Even without the Suzy present, although the system will not boot, you can check the Hayato is at least in some way alive.
Connecting 5V power to it (through the Safe Boot or Power Circuit method), the Hayato should output around 2V on pin 60 (bottom left corner right by the crystal) when the crystal is connected and working and the crystal pin 59 should start to oscillate and you will see a 16MHz sine wave on pin 59.
A dead Hayato usually does not output the 2V on the pin 60 because in order for the crystal to oscillate the Hayato actually outputs a weak 5V pull up on pin 59. This cannot be seen when a crystal is installed as it oscillates over that pin.
Crystal Oscillation Pin Resistance
A dead Hayato with no power applied will normally have a resistance from pin 60 to ground, and pin 59 to ground, of less than 500k, usually around 360k.
A working Hayato has a resistance of 600k (0.6M ohm) or more.
Confirm 5V On Pin 59
To confirm it is not the crystal or resistor at fault, remove the crystal and R10 from the circuit, and measure pins 59 with power applied.
The Hayato should output a solid 5V DC on the pin 59.
If you see no 5V on that pin, then your Hayato is pretty much confirmed dead.
Confirming Crystal Is Good
If the 5V is present on pin 59 with no crystal or resistor, reinstall the R10 resistor (should be around 10M ohm), and measure again with the power applied but still no crystal installed. Now pins 59 and 60 should have the same voltage, of around 2V DC.
If it does, and reinstall the crystal, power back on, and you should see 16MHz oscillation on pin 59. If you do not, your crystal is bad.
Confirming Hayato Audio Out
Once the Hayato has the 16MHz crystal oscillator, the next thing it will do without any game, screen or Suzy present, is to output a 5V 50/50 PWM 1MHz square wave on pin 10 (High Right) and pin 17 (High Left), and a shorter 5V 10/90 PWM 1MHz square wave on pin 11 (Low Right) and pin 24 (Low Left).
If you see these pulses its a good sign the Hayato is alive and kicking.
Confirm Address Pins High
The only other signs of live you can check on the Hayato without a working Suzy is the address pins 30 to 38, excluding 31, will go high to 5V for about 1 second during first power up. Then they drop back to 0V if no Suzy is detected.
From this test (without a Suzy attached to the board), I would conclude the Suzy is similar to the MPU on the Game Gear in that the Suzy is likely responsible for driving the address lines and reading in the data from the game.
We already know the Suzy generates pixel data, so is possible it also reads and processes the entire game, leaving the Hayato up to driving the LCD and running audio, but this is not yet confirmed.