Scratching my god given Head that is filled with the most valuable and expensive electrical knowledge! I read all above and have concluded that even the worst foreign engineer did not design a device that is sensitive to 0.1 volt!!! or 0.2 volts... NO WHERE!
Keep in mind the voltages you have shown are very arguable from a perspective as to how they were read with a meter and accuracy of the meter.
If you were reading these voltage in "open circuit" not connected to the circuit while charging- Theres the answer.
In my power supply and charging knowledge and experience- I am betting that your devices when connected were sending the power supply to the rail! (thats what we call not enough). the solar panel was falling on its face and couldn't deliver. Thats exactly what i found with solar panels I bought. they didn't deliver the units of energy as specified. not even close.
The regulator you bought was able to switch on/off to keep the applied voltage up. thats why it works. I'm betting if i looked at it on a scope, its pulsing the output to give the power supply time to catch up.
The other power supply or "chargers" were to small also?
There is considerable junk on the market for chargers.. seen them, modified them, thru them away. As spiderman said- theres bad designs out there that replicate junk voltage signals. Buy quality stuff. even for your phone.