Wired incorrectly, while letting the fan use 2 return paths for neutral wouldn't be the end of the world, it is technically against code and unprofessional. The only way to wire the scenario with power first coming to the switch box is to also wire the fan to that first switch box. You would need 12/3 from the fan to the 1st switch box. Your other 2 switches controlling the fan would then be wired as a 3 way/ four way switch loop.

Wiring needed
1st switch box:
-12/2 from panel
-12/3 to fan
-12/3 to 4 way switch continuing to other 3 way switch.

The light will then be using the red wire, connect it to the common terminal of the switch. Connect the white of the 12/3 heading to the other switches to the black incoming hot. At the 4 way switch, connect the white to the white heading to the last 3 way switch. At the last 3 way switch connect the white wire to the common terminal and the black and red to the other 2 terminals. At the 4 way you'll have 2 pairs of black and red to connect to the 2 pairs of terminals.

The fan switch is easy, just connect a black to that same incoming hot wire that is tied to the white heading off to the other switches. Connect the black heading to the fan to the other terminal. The incoming power white wire will connect to the white wire heading to the fan.

Yes it is complicated but it is the correct way to wire this. It simply comes down to the fan/ light not being allowed to use 2 different white/ neutral paths back to the source wire. The code only allows 1/0 and larger wires to be used in parallel, #12 wire is a lot smaller than 1/0.

spence is right.