If you buy one of the square solenoids (slightly more modern design) you can easily get them apart to look at the contacts. It is only 4 rivets to drill out, then they can be replaced with nits and screws. The type Bizzard linked to have the insulated terminal end swaged into place, so is more difficult to get apart and far more difficult to re-swage. The square type is the one shown in the manual I found and may well be the one Lister agents would supply. I don't think they are particularly less reliable than the round type.
There is not much inside either type apart from a single coil, a contact bar, and two contact pads that are part of the terminal studs, so I too suspect the fault may be else where, like a terminal with a poor connection (crimp) to the cable, or, as said, a problem inside the starter. After all, those solenoids are designed to pass the full petrol engine starting current, but you are only passing the current the starter's internal solenoid coils require, which is likely to be far less. Sorry, I do not have any values.
I wonder if internal modern solder joints are failing, but if so pushing the "button" should still operate the starter, so the fact it does not suggests dirty/burned contacts or an open circuit elsewhere.
Some square ones have a little rubber (often red) push diaphragm to bring the contacts together, others do not.