Does the level in the fuel tank matter? If so, could be a pinhole or fracture in the pickup side pulling air. It could also be sucking air when low on gas and you stop hard, sloshing the gasoline away from the pickup.

Its either a hole in the in-tank fuel line or the variable speed sensor. Both can cause the problem you're having. I had an 03 Road King doing the same turned out I had a pinhole in the fuel line that's in the gas tank. But I cleaned the VSS and it was real dirty too.

Start with cleaning the Idle Air Control passage "IAC" first. Do a search ad there are treads about this.

I pulled these pics from another thread and although the location is slightly different on some bikes, its somewhere close to the starter. I had a little bit of a hassle getting to mine on my 03 King due to how close it was to the starter. It was tough to get it fed through the tight spot it was located in but it may be different on yours.
There will definitely be some metal on it when you pull it off so don't be surprised. As long as its not a lot its fine. Wipe it off and reinstall or replace it altogether.

Check the ignition switch....Pull it and clean the contacts.

Crank Position Sensor may be going bad.

i had the same issue, the root of it all was a low battery......for me.