Why does the crystal cause the doubly-inverted thing to change?
The first step is to clear up a misunderstanding.
You refer to crystals being connected to a voltage rail.  That would be the case for a crystal oscillator.  That's a module that includes the crystal and the associated circuitry to make an oscillator out of it.
FrancoVS provided a link to the Wikipedia crystal oscillator page.
On that page, you will find this image:

To the left there is a crystal.  It's a piece of quartz in a housing with a couple of pins bonded to the crystal.
To the right is a crystal oscillator.  It includes the piece of quartz, but also an IC to provide the amplification and buffering needed to actually generate a signal from the crystal.
Wikipedia also provides this image of the insides of a crystal oscillator:

To the left in that picture is the crystal (the round thing.) To the right is an IC that provides amplification to make the oscillator.  A crystal oscillator has three connections: Power, ground, and output.
Your circuit assumes a simple crystal.  The item shown as Q1 in your schematic is simply a piece of quartz in a box with pins.  It needs additional circuitry to make an oscillator.
The 7404 provides the amplifier that you need to make an oscillator with a frequency controlled by the crystal.
In this case, look at the 7404 buffers as inverting amplifiers.  Two of them in series is a non-inverting amplifier.
To get oscillation, you need positive feedback.  The circuit you show provides positive feedback through the crystal - but only at the resonant frequency of the crystal.
The first two 7404 inverters (D1(LN1)A and D1(LN1)B) are the oscillator that generate the signal, with Q1 setting the frequency.
The third inverter (D1(LN1)C) provides a buffer between the oscillator and the rest of the circuit.
D1(LN1)C has two jobs:

It sharpens the oscillator output from a sine wave into a square wave.
It prevents the load from other parts of the circuit from interfering with the operation of the oscillator.

R1 is probably a little bit tricky.  It influences the start up behaviour of the oscillator.  The correct value will depend somewhat on the exact 7404 model that you use and the crystal itself.  Some combinations of crystal and inverter will oscillate without it, some will need a particular resistance value to make the oscillator start reliably.