I too think the starter switch may be the issue. Remove the switch (shiny piece on top of starter) and then jumper the battery cable to the copper disk on the starter unit. The starter should spin if it is alright.
3 more bolts to remove and you can try the starter out of the car and see what is happening. You will have to remove it anyway since it is not working.
When you tried to jump the starter with the battery cable, did you notice if the copper strip in the starter switch and/or the button that the copper strip contacts is corroded? I have had a situation where they have been corroded and the starter would not turn until the corrosion was cleaned up.
A 6 volt battery in good condition should show about 6.3 to 6.4 volts, especially right after the charger has been on it. You'll get a higher reading from the charger just being on it, but if you turn the lights on for 15 seconds, then turn them off and read the voltage, you should have 6.3 or close to it.
Try hand cranking the engine to be sure the starter drive isn't jammed in the flywheel teeth.
I think the battery may be toast. 5.99 dropping to 5.75 Volts for a battery may be too little amps left to run the starter. as Tom said, you should have 6.3 volts on a fully charged battery. Check the voltage drop when you depress the starter switch.