I always had the misconception that when people say ions conduct electricity, what they actually mean is a connection is made between the cathode and the anode by the ions and electrons can flow from the cathode to the anode through this connection.
However, I recently learned that ions conduct electricity because chemical reactions are happening around the electrodes.
Taking the example of the electrolysis of sodium chloride water solution. At the cathode, $Na^+$ captures electron ($Na^+ + e^- \rightarrow Na$); at the anode, $Cl^-$ releases electron ($2Cl^- \rightarrow Cl_2 + 2e^-$). And this is because the battery takes electrons from the anode and feeds electrons to the cathode, and positively charged anode attracts anions and negatively charged cathode attracts cations. That is to say electrons do not flow from cathode to anode but are given to and taken from the ions.
I still feel confused after knowing this explanation because it seems to me that the anode and the cathode do not need to be connected by aqueous electrolytes to "conduct electricity". Suppose the cathode and the anode are placed into different solutions in different containers, anode still attracts anions and cathode still attracts cations. So that there should still be electricity flowing through the circuit.
Some people say that a "closed circuit" is required for ions to conduct electricity. If that is the case then what am I missing here?