The sludge trap is inside the crankshaft so it would need a total stripdown to clean it out. I've never done this on my A7 as it was bought in running condition, about 12 years ago. Whether you strip the engine at this stage is up to you. If you think the bike was neglected and poorly maintained, with infrequent oil changes it would be the safest policy. If it was well looked after and does not show signs of bodging or butchery you could get away with running without stripping. You should still do a bit of servicing, things like valve clearances, points gap, ignition timing, check the clutch, primary chain, brakes etc. It would be a good idea to remove the primary chain cover as there may be a pile of broken rollers in there. It is also worth stripping off the clutch plates and filing burrs off the clutch plate tenons. Similarly, if the surfaces of the clutch basket are grooved by the plate tenons, they can be filed smooth.
I'm not sure about using flushing oil on an engine without proper filtration. It might be safer to just remove the oil tank and give that a good clean out with kerosene or some such. You can always change the oil again after your first few hundred miles if you think there is sludge build up. One place to check for this would be the rocker gear. If the rockers are thickly coated in sludge then a full stripdown would be best. These bikes do tend to lose gearbox oil as there are no seals on the gear lever or kickstart lever shafts.
As you have discovered, the bike will run without any wiring apart from the magneto HT leads. The kill button is a useful safety feature in case the throttle ever sticks open. At a show last weekend I saw an A7 in scrambles trim which is used on the road but has no lights, dynamo or wiring, just a rubber bulb horn.