I purchased a used set of wheels for an old Schwinn. I'm looking to replace the hub because when I put the old Schwinn front freewheel system cartridge on it, the spacer rubs quite badly against the inside of the cartridge. There are no lock nuts, cones or anything else to be seen. I'd like to remove the hub and replace it. Thanks!
You don't.
That's a hub for a freewheel, which threads on and the thread is part of the hub. All the pawls are inside the freewheel which is not supposed to be serviceable, and is replaced when you buy new cogs.
The modern wheel has a hub, a separate freehub, and a cassette which simply slides onto the freehub.
To change the hub, you'll have to undo all the spokes and assemble the wheel with a different hub, likely different length spokes, and the same rim.
A better solution might be to sell this vintage wheel as-is, and buy a more modern replacement wheel with a cassette. However you've mentioned a "Front Freewheel System" which is particularly vintage, and I suspect you're trying for a period-correct bike.
from https://en.wikipedia.org/wiki/Front_freewheel
This setup is unusual - the front chainrings can freewheel when coasting, so the chain keeps moving and you can change gears while coasting. To keep the chain moving, the rear cluster does NOT have a freewheel or other escapement. The only reason its threaded onto the hub is to facilitate replacement when the teeth are worn.
I fear you might have problems finding a replacement cluster.
