Well, look at it this way...
You get a flow of charge, whether ions or electrons, when there is a charge imbalance, i.e., there is an excess or shortage of charge of one type or another at one part of the circuit compared to another. The current flows to neutralize this imbalance. This, we call potential difference or voltage.
Electrolytes are electrically neutral. There is exactly the same amount of net positive charge as there is net negative charge though the actual number of ions may vary. As a result, there cannot be a potential difference within a free flowing electrolyte solution. We can have a potential difference, if the solution isn't free flowing.
As in the above picture, from https://www.revolvy.com/main/index.php?s=Polymer%20electrolyte%20membrane%20electrolysis you can put in a semi-permeable membrane between the electrodes of your cell. Now, if you pump in electrolyte solution on one side, the smaller ions can migrate through to the other side, the larger ions can't. So you've got a difference in potential, a tiny one to be sure, but you do have it. In fact, you have what could be called an electrolytic diode, and given an appropriate stimulus, you could get a tiny current across the electrodes.
Except that you don't want electrical current; you want ionic flows. In order to do that, we pull the net charged liquid out with a pump and send it towards an oppositely charged electrode, which will, hopefully give you a miniscule current, assuming that the ions don't discharge in air or within the pipeline.
In other words, you can do it, but the cost would be a net negative energy transfer, i.e., you would be putting in maybe a thousand times the energy you will recover.
Now, if you replace the liquid with plasma, you can form a plasma channel, which has far less resistance to transmit charges from a static generator. Here you have the costs of generating the plasma in the first place, in addition to the fact that convection is several orders of magnitude worse at transmitting energy than conduction. On the other hand, there will be less electrical resistance.
In short, you can do it if you really want to, but it isn't very practical.