This is one of those topics, like what is better AC or DC, that can have people arguing for ever. However, despite my better judgement, I'm going to weight in.
Voltage Causes Current
Let us consider the two primary methods of creating a voltage.
1. A BATTERY
A battery is one of the most miss-understood devices we use. The classic boilerplate "why is the sky blue?" answer to how a batter works is....
"Batteries have three parts, an anode (-), a cathode (+), and the electrolyte. The cathode and anode (the positive and negative sides at either end of a traditional battery) are hooked up to an electrical circuit. The chemical reactions in the battery causes a build up of electrons at the anode."
We have passed that at the answer to "How to batteries work?" since the thing was invented and it is generally accepted as a "that makes sense" answer, and people leave it at that. People think of it like a capacitor, that the anode and cathode are like capacitor plates with a charge built up on them.
Unfortunately, it is also totally wrong.
If that were true, two things you know would not be true. First, if the description above were true, a bigger battery would mean more chemical reaction and therefor more voltage. But we know that's not true, a AAA alkaline battery has the same voltage as a D-Cell. Second, if a chemical reaction is causing the voltage, an ideal battery (no current leakage) could not hold it's charge forever if left unconnected. The chemical reaction would, at some point, "burn" itself out.
The truth is a battery generates voltage without a reaction taking place. The reaction happens when you allow current to flow between the terminals.
So what is REALLY going on.
Batteries work because of a lesser known physical effect called Electronegativity.
"Electronegativity, symbol χ, is a chemical property that describes the tendency of an atom to attract electrons (or electron density) towards itself. An atom's electronegativity is affected by both its atomic number and the distance at which its valence electrons reside from the charged nucleus. The higher the associated electronegativity number, the more an element or compound attracts electrons towards it."
What does that mean? It means when you bring two dissimilar elements together in the presence of an electrolyte, a material that facilitates the motion of ions between the metals, there is a force generated that wants to pull electrons from one side to the other. We call that force VOLTAGE.
When you build that you create a Galvanic cell.
"In a Galvanic cell, the metal atoms of one half-cell are able to induce reduction of the metal cations of the other half-cell; conversely stated, the metal cations of one half-cell are able to oxidize the metal atoms of the other half-cell. When metal B has a greater electronegativity than metal A, then metal B tends to steal electrons from metal A (that is, metal B tends to oxidize metal A), thus favoring one direction of the reaction:"
When you connect the terminals together via a conductor, the voltage pulls electrons from one metal to the other through that conductor. This allows ions to move through the electrolyte and the chemical reaction to take place.
But again, to be clear, when disconnected, nothing is moving. There is no current, no chemical reaction and no build up of "charge". But the voltage is still there.
2. ELECTROMAGNETIC INDUCTION
Consider the circuit below.
simulate this circuit – Schematic created using CircuitLab
OK so if you know your stuff, you know that there is 100V AC appearing at the outputs of that transformer. That's easy enough to understand, right?
But how much current is flowing?
The answer is...... NONE.
You know by conservation of energy, that since the outputs are disconnected no power is being removed from the output of the transformer and therefore no power is being inserted into the input either. Again, as a consequence, no power is being removed from the generator either. Current in the above circuit is zero on both sides. There is no motion of charges.
So how on earth is there a voltage on the right side?
The truth is the rotating magnets induce an electric field in the coils of the generator which is propagated through the conductors at the speed of light. That moving electric field generates a magnetic field in the transformer which in turn generates the opposite electric field in the secondary.
That all happens without any charge movement.
In the primary the transformer effectively produces a back-emf equal to the emf of the generator. Since the Voltages are the same no current can flow through the wires.
When you attach a load to the secondary, the voltage generates a current which causes it to collapse some of the electric field. The back emf on the primary collapses in sympathy. That unbalances the voltages in the primary and current flows from the generator.
A FINAL DEMONSTRAION
Consider a DC motor. OK we all know you attach a voltage to the motor the motor gets all "excited" and starts turning. Current flows through the motor and creates torque.
But then that odd thing happens called "Back EMF".
Here is your final proof that voltage is not caused by current.
You KNOW your current is "going in" the RED LUG of the motor and yet miraculously this voltage is appearing in the wrong direction. Charge can't be building up on the coil because you know the electrons are going the other way.
Voltage is, a thing to itself. It needs no circuit current.
The truth of the "matter" is it's all theory and conjecture to try to build a workable model to describe an observational effect.
As with all theories, the more answers you give the more questions get asked. Further each theory has it's own set of believers and disbelievers. Theories and models also change with time. That's why we call them theories.
It's like peeling an onion. The deeper we go, the weirder it gets.
However, at the macroscopic level we work at, you open the switch, the current disappears but the voltage remains. At the end of the day, that's all that really matters in this forum.
How it physically gets there is not really of any great consequence unless perhaps you are building a quantum computer on your breadboard.