TLDR: Seek out 14/4 cable, or run conduit and individual wires. The neutral must be in the same cable or conduit with its partner hots. Neutrals and grounds cannot be borrowed (for your application). Ground cannot be used as a conductor. Or, use a smart remote.
If you're an electronics guy, Code is full of weird idioms that don't make sense until you get deeper into the craft of it.
Neutral is not Ground
It's common in DC electronics to use "ground" as both RF shield and current return.
Code electrical is wired the same as if it were a fully isolated system with equipment ground/shield only as a safety device. To keep voltages from floating/rattling, it is bonded to earth in one location. Only hots or neutral may return current, never ground unless you want a GFCI trip.
Because many people must service a circuit, colors are standardized: Neutrals are always white or gray; grounds are always green, green/yellow or bare. Any loose ground wire (not in one of the unusual, grandfathered applications) will be added to the ground bundle by the first competent electrician who sees it attached to anything else.
You can't use ground as a conductor.
That bare wire can only be ground except in a very narrow list of exceptions. All of which involve grandfathering old work. So forget it. Ground as a conductor is not gonna happen.
Only when you are retrofitting ground to old (1970s> work) can you steal ground from another circuit. When doing new work, you must "do it right".
You especially can't steal neutral
Your original scheme, and ThreePhaseEel's modified scheme with 14/3, involves stealing neutral from the light circuit. Nope. You need to put your neutral in with your partnered conductors. Your best bet is to find 14/4 or 14/2/2 cable. It's a bit harder to find but is a Code requirement for what you are doing. Either will have 4 conductors and a ground wire. Alternately, you can run it in conduit and use single conductors, as many as you need/like.
Your motor control needs one neutral and three hots in a single current loop. All of these conductors must be inside the same cable (or raceway/conduit).
It is absolutely not permitted to share a neutral wire out of a different cable. That's because in any cable, the current of all the conductors must be equal/balanced (or to be more precise, sum up to zero when flow direction is accounted for). In this way, the magnetic fields from all the wires cancel each other out. Otherwise, a magnetic field forms around the cable and particularly between the unbalanced cables, and that will inductively heat anything with the misfortune to be metal. Remember, we're dealing with AC here. This is a big deal. This sort of thing is why AC motors need laminated field windings, and DC motors do not, even though the motors are identical in every other respect.
A good litmus test of this principle is to imagine what happens if you put a GFCI on the device/circuit in question. If hot and neutral current are unequal, it will trip. If neutrals are borrowed or ground is used as a conductor, it will trip. Every Code installation should pass a GFCI test.
Like I say, a lot of Code stuff seems weird... or even dumb. "But you can do that!" Actually, get into the gory details and you find out when people did that, houses burned down. The system is intensely designed to save lives and be maintainable by any licensed electrician or Code-respecting hobbyist without obsessively googling every product they might find or weird installation they might uncover.
Plan B: use a smart module
A much better way to do this thing is to use a smart device which is made to control fans and lights, and a partner "remote" that sits up in the fan cowling. The smart control communicates with the remote using fewer wires. In a perfect world, it uses radio or inducts signal onto the power line, in which case you only need to bring always-hot and neutral to the switch and fan location.
The remote sits between the hot+neutral and the fan, switching (dimming?) the light, and selecting the various fan speeds.
I assure you this product exists, so search for it before you launch a Kickstarter.