I noticed the shift cable. I would route it along the frame
That could be a contributing factor, especially is the housing is limiting the movement of the derailleur.
Another factor is the shifter itself. What you have isn't stock. Is that a Suntour? Is it a click shift? If so, then you've got to get precise, especially with the old shifters. Hopefully it's friction. Regardless, let's move on with the adjustment advice.
We'll assume the housing is allowing full movement, the dropouts are straight, and the derailleur is straight.
Your first adjustment needs to be the cable. Assume the tension is wrong. Break it loose at the derailleur and turn the adjustment stops all the way in at the derailleur and if there is one up by the shifter, up there too. Those are used for fine tuning your cable tension, since that is what is used to actually move the derailleur. Remember what I said about precise? These are what gets those there. So, once you've released it, and turned in the stops, pull it though again hand tight, then tighten down the cable.
The next adjustment is the high gear stop. That's the smallest cog. There is a screw on the derailleur, and I believe it's the one inserted into the face, that stops the cage's movement before it pulls the chain farther than the highest (smallest) cog. Turn that until the chain is running straight on the cog, and there is no pulling (no clicking sound) from the derailleur. Now, shift down to 3rd gear, then back up to 5th. The chain should fall from 4th down to 5th, then go no farther. If it struggles to fall into 5th, then loosen that screw a little while you're pedaling until it falls and is quiet.
Next is the low gear (largest cog) stop. Do the same thing as above, except with the other screw. This one is important because it has to be loose enough to allow it to get into 1st gear, but tight enough to prevent the chain from falling behind the freewheel, or before the derailleur rubs the spoke protector. (This is where checking for verticalness and twisting will pay off.)
After that, you need to fool with the cable tension. Tight enough to allow smooth, quick drops into low gear, but loose enough to allow smooth quick moves into higher gears. This is just an art, and you have to play with it. If it's friction it will be a lot more forgiving because you'll have to find the spot where the bike is quiet while you're riding. If it's a click shift you have to find the balance in the tension to find the sweet spot. The problem is: In this old 60's technology, there may not be a quiet sweet spot. Or it may be very difficult to find.
Tips: Lube the chain, lube the derailleur, lube the freewheel. I use chain oil for the chain, and silicon spray for the freewheel and derailleur. (Spray the actual mechanical guts of the freewheel. The chain will take care of the cogs.
Hope this helps.