You only have always-on power at the first switch (the 3-way by the door). At the fan/light, you only have switched power, controlled by the two 3-way switches. So, you need to run a 14/2 connected to the power coming from the panel (at the switch by the door) to the two closet switches to supply them with power.
You need two wires to have an electrical circuit - "hot" (black in house wiring) and "neutral" or "return" (white in house wiring). When you wire a 3-way switch, you need an additional wire connecting the two switches: The neutral wire is always connected, and the switches choose which of the two other wires to make the hot leg of the circuit. When both switches are connected to the same wire, the load gets power (the light/fan comes on), when they are connected to different wires, the load is switched off because there isn't a complete circuit.
You could have an "always on" circuit at the light/fan, but you would need to run 14/4 - four conductors - between the two 3-way switches - neutral, plus the always-on hot lead, plus the two switched wires. Then, from the second 3-way switch (the one connected to the light/fan), you would run 14/3 to the fixture, with neutral (white), hot (black) and switched (red). At the fixture, you would connect neutral and switched to the light/fan to supply it with switched power, and you'd have the neutral and hot leads to continue to other uses, such as your closet lights.
That's not the way I would do it, though: As I said, run a 14/2 to supply your closet lights, tapping off the mains at the 3-way by the door. That avoids having to use 14/4 altogether, and you only need 14/2 going from the second 3-way to the fixture - neutral and a (switched) hot lead.
Referring to your diagram, instead of having a 14/2 going from the ceiling fixture to closet #2, have a 14/2 going from closet #1 to the switch by the door. There, as I said, connect the new 14/2 to the wires coming from the panel for power. No other changes are required from your plan.
Actually, you don't need two 14/3s, you only need one 14/3, plus a 14/2, going between the two 3-way switches because you don't need two neutral wires:
In this diagram, the blue lines represent the neutral (white) wires - which wouldn't show up well if they were white lines :) The selection of whether to use red or black to connect the light and fan hot leads between the inside 3-way switches and the fixture is arbitrary - pick one that works. As you can see, the white wire in the 14/3 between the 3-way switches is the neutral. Assignment of the other four wires is also somewhat arbitrary, just be sure you have the switches connected in matched pairs.
I think you have mis-identified the "hot" wire. Do you have a multimeter? Assuming your box is actually grounded, carefully measure the voltage between the red, black and white wires to the box. If only one has 120 volts on it, then that is your hot wire. If 2 wires have 120 volts, then someone did a bad thing and wired the switches to break the neutral.
Another way to identify which wire is hot is to pull down the light fixtures and examine the wiring. Don't disconnect anything without taking notes about how it was hooked up first!
Best Answer
The driver circuitry May have been damaged or had cold solder joints that are failing.
The led dimmed or not should be almost immediate.
I have had drivers that had cold solder joints on very high end fixtures that caused the fixture to take longer to light to start with and a while later it would not light at all as these fixtures were almost 500 ea I figured out the problem and it a bad solder joint (I don’t have to mention where the fixtures were made but it was not the US)
If you still have a warranty I would recommend trying that route because if any LED won’t light in under a second something is wrong.
I would also recommend getting DLC approved lamps or fixtures DLC is design Light Consortium and they usually have a 5 year warranty (some 3 year) . All the ones I have seen also have UL listing (I have to have this for industrial or another nationally recognized 3rd party facility).
Since the lights are taking more than a second there is a problem it could be a loose wire can be a possibility. My experience has been bad drivers on almost 100% of my failures.