Flickering light at outlet tester
Everything you have observed points at a bad connection with high resistance. When the load is small (voltmeter or neon test lamp), the measured voltage is high, and when there is a larger load, the voltage drops. LED and CFL lights may flicker because they can start with very little current and they have capacitors in their power supplies enabling momentary running on the tiny available power, but an incandescent bulb will stay dark because it never gets hot enough to be visibly lit.
You will need to find the bad junction. One way to do this is, while there is a load (the sconce lights), to put a voltmeter across two different points on the same wire (both hot or both neutral). Good wiring should see no more than a few volts, and usually much less; a bad connection will show somewhere up to the full 120 volts. If you see a large voltage, the bad junction is somewhere between those points, and you can then test further within that span, or inspect all the joints. You may need extensions for your meter's leads to be able to reach.