Does this hub not have an oil port ?

You want to be very careful when you take a 3 speed hub apart as to not have a million little pieces all over the place... if they can be oiled that is usually all they need and your shifter may also be a little out of adjustment. With an older Shimano the red N needs to be centered in the window.

With old SA hubs I don't use a visual check on the spindle chain unless I know it is the right length as there are several sizes and who knows if the spindle is stock... it is better to set the cable tension while it is in 3rd gear.

There should be no pull on the spindle in 3rd and the smallest amount of slack that should get taken up when you apply a light touch to the trigger (without actually shifting) and the bike should shift up and down (on the stand) as you turn the cranks at low speed.

If it slips into neutral while in 3rd the cable is too tight and if it slips into neutral while in 2nd it is too loose.

SA 3 speed hubs do not like to be shifted under hard loads as it is not a perfectly synchronized transmission.