Read your original post where you described the axle as a custom 50" long. You said that the 40" long center section is 1" OD that's keyed the full length of itself (40"), and that 5" of each end is stepped down to 5/8" that's threaded the full 5" length on each.
I've never seen a stepdown from 1" that was any size other than 3/4", but you say it's a custom axle, so I must believe it's the 5/8" you stated. Note the size of the stepdown hub in the link you just posted, 1" stepped down to 3/4", pretty standard. That would slide onto a 1" shaft partway, then the stepdown in it would bump up against the larger 1" and stop. You would then tighten a 3/4" axle nut to the hub and you're done.
That would still work for you with your 5/8" ends because of the keyway in the 1" portion, it would lock the axle and hub together. That's the part that's supposed to take all the force, the small end doesn't really matter.
Your concerns are the wheel bearings of the Harbor Freight wheels riding on the threads, if that's an acceptable fit or not, and the lack of a keyway in the threaded 5/8" section of each end of the axle.
Return the wheels to Harbor Freight, get your money back. You can't use any wheel with bearings in it, the wheel has to lock to the shaft, a bearing would make it spin.
Get the hubs you listed above if that's what you want, they will go onto your axle. You'll need a 5/8" ID washer that has a larger OD than 1" to go on next, a common washer would work. Then just run a 5/8" axle nut tight to the washer, and that part's done. Note that axles have fine thread on them, not common coarse threads, and an "axle nut" is just a nut with that fine thread and a nylon insert that keeps it from vibrating loose.
You'll then need to get wheels (rims) that fit a 4-lugs on a 4" bolt circle, known as 4 x 4 (see your listing above). Get tires for the wheels (rims) and you're done.