Starter or solenoid not working?

Jump across the solenoid with a heavy piece of metal or cable and see if it cranks.

BW 

 That's what the hand crank is for....

Even when I was in HS, I always parked it on a hill. When It needed a tuneup, I used a long hill. Yes, I am lazy!

These days I fix the problems. Yet, I still almost always park pointed out & down hill if possible. Old habits die hard.

Smart move to ensure all connections are clean.  Nothing will fix this if the connections are "insulated" with crud.  Agree with Seabee.  Have you tried to jump across the solenoid to bypass (sending current directly to the starter without switch input)?

I remember similar, by ear, diagnoses to differentiate between solenoid and starter but the hum is a new one.  You're sure it's not the starter spinning without engaging the bendix?

The hammer on the starter has been "hit and miss" with me as it needs to be able to jostle the armature to make intermittent contact with the brushes to get the starter to turn (almost a last-ditch-effort when all else fails).   I think if there's enough wear on those parts, even a series of solid smacks won't help. 

My go-to was always park on a hill, "just in case."

  “ The solenoid does not affect charging.”  Not normally, but with a rat’s nest like that, who can be sure?

OK, a couple of things, the first is those connections are not clean. The entire connection should be bright & shiny like the new cable is. Use a wire brush or sandpaper on all the metal connectors & the posts on the solenoid.

Those two smaller wires coming out of the black plastic connectors with the embossed "16" on them, are they fusible links from the regulator perhaps? If not where do they go? What the heck do they feed?

This is a 12 volt system right? I see a 12 volt battery & an alternator but alternators come in both 6 & 12 volts varieties. Yes, I have seen where a 12 volt battery was installed in a 6 volt system, it does not charge very well.

The red wire coming from the alternator has a yellow butt connector, are either of the wires coming out of the butt connector loose? Heck, are any of the connections to the alternator loose or corroded?

That new positive cable still looks kind of small to me. It might work for a 12 volt system in warmer climes. What size is it? What size is the cable from the solenoid to the starter? The blue ground cable from the battery to the frame or engine block is what size? It looks to be one size smaller than the positive cable, it should not be. Are the last two cables in need of replacement? Are they frayed or corroded? All the cables from the battery to the starter including the ground cable should be the same size, otherwise the smallest cable is limiting the amperage that the entire system can carry.

Enough questions for now. IHTH, Cpt Logger.