Can't get lowest gear and derailleur shifts on its own

Question

I have a bike with downtube friction shifters and a 5 speed freewheel. When I bought the bike a while ago the shifters weren't connected so I did it myself. At this point it all worked fine. I took it to the local bike shop to check out what I had done, as I had never done it before. He did some adjustments (I don't know exactly what) and ever since I have had trouble with getting the lowest gear (biggest cog) and the derailleur shifts automatically back down to the smallest cog.
I have checked:

the cable - it is tight
the limit screws - they are fine
the shifter - even when tightened I have the same problem

Today I notice three things:

the cable binder bolt on the rear derailleur is loose, so it spins on its attachment to the derailleur. I assume this is a problem.
the cable is frayed after only a couple of months use
in order to move gears I have to put a huge amount of pressure on the shifter and it only uses about one third of the possible span of movement. When I set it up in the first place the lowest gear and highest gear were almost 180 degrees apart on the lever (if that makes sense).

I now need to replace the cable but how can I fix the derailleur issue? Does it just need replacing? I put a photo showing the position of the derailleur in 1st gear to check if the chain is the correct length.
I don't really want to go to the bike shop again as he isn't the friendliest person....
Many thanks for your help.
Gill

! This is the shifter position when it is on the big cog.

Argenti Apparatus · Answer

Sounds like the cable is pulling through the binder bolt on the derailleur, so that the shifter cannot pull the derailleur inboard onto the largest sprocket.
The shifter cable needs replacing. Check that the housing is properly located in all the frame stops. It would be good to put a ferrule on the end of the shift cable also.
It's possible that the derailleur lower limit screw is set to high and is preventing the derailleur moving inboard enough to engage the largest sprocket properly. You can check this by manually pedaling and carefully pushing the derailleur towards the largest sprocket.

David D · Answer

We know it worked once, but we don't know what was changed, so it's time to go back to basics.
Auto shifting
The rear derailleur has a spring that wants to move the chain to the smallest cog.
The shift lever should have enough friction to keep the derailleur spring from going where it wants but not be so tight that it won't move.
The first thing to check is that the wing nut is tight.
If the wing nut is tight and there is still auto shifting then it's time to check how the levers for correct assembly.
My guess is that you have shift levers that look something like this:

Check that all the washers are there and in the right place.
Low gear
There are at least three main things that can keep you from getting low gear.

Limit screws - you have checked them and they are fine. Just to make sure, can you move the derailleur by hand and get the chain to go into low gear? If so then we know it's not the limit screws.
Cable tension - with the chain in the smallest cog and the shift lever all the way down the cable should have no slack.
Derailleur alignment - here's a picture:

Picture from SOS Hanger

Can't get lowest gear and derailleur shifts on its own

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