I do not remember what AVR does. In recent years I have used mostly ST microcontrollers and all of them had Schmitt-trigger inputs on the GPIOs. Obviously, for any microcontroller the datasheet must be checked when a schematic is designed.
The output of the noise amplifier is quick aperiodic oscillation centered on a value that is not a valid logic level. There is no guarantee about how fast will be the transition times between the logic low and high levels.
For a gate with Schmitt-trigger input, it does not matter whether the inputs have valid logic levels or not.
On the other hand, for any standard CMOS logic gate, there is a maximum time during which the input is allowed to be not valid. If that maximum time is exceeded, then there are no guarantees about the power consumption of the gate and there are no guarantees that latch-up (an internal thyristor short-circuiting the power supply) cannot happen (in normal operation, only one of the two transistors of a CMOS inverter is on, while the other is off, so only leakage currents are consumed; when the input is invalid, both transistors are on and an unpredictable current passes through them; the time while the input is invalid must be limited, so that the transistors will not overheat).
Of course you may be lucky and nothing wrong will happen when a certain CMOS gate will have an invalid logic level at input. However, such a schematic would never be acceptable for a product made in large quantities, because it is unpredictable how many will fail and when they will fail. Moreover, if such a schematic would be used with a power supply that does not have overcurrent protection, it could even be a safety hazard.
The older bipolar gates like 74ALS04 do not have this problem, but their more modern replacements, like 74HC04 or 74LVC04, must be used as in their specifications, they are much less forgiving of being used in undocumented ways.