In this answer, I'll discuss conducting electricity in electrolytes for three different scenarios, an electrolytic cell, a Voltaic cell, and an electrophoresis setup.
Electrolytic cell
Let's take the synthesis of elemental sodium from a sodium chloride melt as example. The net reaction is:
$$\ce{NaCl(l) -> Na(s) + 1/2 Cl2(g)}$$
The electrical power source provides electrons to the cathode, where they reduce sodium cations that traveled from the melt. Chloride anions from the melt travel to the anode, and turns into chlorine gas, providing electrons to the anode going back to the electrical power source. "Ion" designates movement, cations travel to the cathode and anions travel to the anode.
I would expect the sodium, once given an electron, to be looking for a way to give that electron back. Would it cause the water to split into hydrogen gas and hydroxide ions? On the chlorine side, would the chlorine atoms simply turn into chlorine gas.
The purpose of this setup is to produce elemental sodium. You run this in the absence of water, and cool down the sodium and store it away from water, air, and other substances that would react with it.
The same textbook chapter I linked to also discusses the electrolysis of brine, salt water. Here, the net reaction is:
$$\ce{2Cl-(aq) + 2H2O -> 2OH-(aq) + H2 + Cl2(g)}$$
Again, chloride ions are oxidized to chlorine gas. Sodium ions, however, do not undergo reduction, even if they are the most concentrated cations in the brine. Instead, water provides hydrogen ions that are reduced.
Voltaic cell
Let's take the Daniell cell as an example. The net reaction is:
$$\ce{Zn(s) + Cu^2+(aq)-> Zn^2+(aq) + Cu(s) }$$
You have the following half-reaction at the anode:
$$\ce{Zn(s) -> Zn^2+(aq) + 2 e-}$$
There is a charge separation. The electron remains on the electrode (making it more negative), and the zinc ion goes into solution (making it more positive). In this case, there are no anions traveling all the way to the anode; instead, the cations travel from anode into the solution.
For completeness, here is the half-reaction at the cathode:
$$\ce{Cu^2+ + 2 e- -> Cu(s)}$$
But what happens once the charges get to the electrodes?
The charge transport between electrode and solution is already accomplished by the half reaction. The role of the ions in solution is the charge transport between the two half cells. This does not involve any redox process.
Source: by Rehua, https://commons.wikimedia.org/wiki/File:Galvanic_cell_labeled.svg
In the Daniell cell, adding zinc ions to the left half cell results in an excess of cations in the solution (excess of positive charge). Removing copper ions from the right cell, on the other hand, results in an excess of anions in the solution (excess of negative charge, to be accurate).
This imbalance is restored by ions (no matter which) traveling through the salt bridge in the appropriate direction. Depending on the mobility of these ions, there might be more cations or more anions moving, but the charge balance is restored in either case.
In this scenario, a sodium ion can help with conducting electricity. However, the sodium never receives an electron (nor does the zinc ion or the sulfate ion in my example). Ions transport charge just by going from A to B. They are charged already, they don't need to "carry" an extra electron. This is different from an electron carrier that shuttles back and forth between two locations, picking up electrons in one and delivering in the other. In the Daniell cell example, you want to avoid zinc ions reaching the other half cell because they would react directly with the copper electrode, without generating a current. One solution to this problem is adding a high concentration of KCl to the salt bridge. The charge imbalance between half cells will be restored by potassium and chloride ions exiting the salt bridge rather than ions going through it from one to the other side.
I keep trying to figure out how current is conducted through an electrolyte but all I can find are incomplete answers.
Your misconception is that you suppose there is a need for electrons to be transported through the solution. However, there is no closed circuit of electrons. The anode acts as a source, and the cathode acts as a sink for the electrons traveling through the wire. This is different from the electrons moved by a generator (electromagnet in a magnetic field), where you do have a closed circuit for the electrons (if the load also lets the electrons go all the way through, like in a incandescent bulb or a resistor, but not like a capacitator or a battery you are charging with the generator).
Electrophoresis
In electrophoresis, you want to separate ions based on charge and shape differences. This is often done in a gel that is easier for small particles to travel through. An electrical field is used to move ions in one direction. Here, you need ions to buffer the solution and to stabilize the sample, and the conductance ideally is as low as possible. Some biochemical methods use bulky ions with low mobility (as in TBE buffer), avoiding the need for buffer recirculation. The samples (protein, DNA) move along the field gradient, contributing a tiny amount to electric conductivity. Most of the conductivity is due to the buffer components, and they contribute to heating the gel, which is often not desirable. The redox reactions at either electrode are a nuisance, but are unavoidable.