They have a energy source, but they actually not using it. They need the energy source to charge the giant capacitor.

But just imagine that after it has been charged to the nominal value the energy source get's broken, and as an emergency measure you just blow up all the conductors and make the two capacitor plates fully isolated. Does the device continue to work? For how long? How do the capacitor discharge?

In their description the particles never hit the plates, so the plates don't discharge. In the real word it's more complicated, but it's an usual assumption for this kind of problems. The real word has these nasty details that reduce efficiency, but they may obscure the main ideas and problems.

If you still are not convinced, just make a hole in both plates and put a isolator tube between the holes. Also, put some isolator around the plates. Make all the plasma only go throw the tube.

Now the calculations are more difficult because it's no longer the electric field of a perfect plate capacitor, but any finite capacitor has problems at the border. If the hole is small enough the using the formula of a plate capacitor is good enough. Let's see a graphic, were the double lines are the insulator and the +++ and --- are the plates:

               entry
  ╔══════════════╦ ╦═══════════════╗
  ║++++++++++++++║ ║+++++++++++++++║
  ║              ║ ║               ║
  ║--------------║ ║---------------║
  ╚══════════════╩ ╩═══════════════╝
                 exit

The efficiency will be abysmally low, because you will have a huge starship with a tiny motor, but the important detail is that the effect will be there. Do the plate discharge?

Moreover, in their calculation they assume that the 80% of the energy is transfer to the kinetic energy of the protons. Where does their calculation change in the model? The efficiency must be also 80%. (Perhaps as efficient, but with a smaller thrust.)

On the Wikipedia page for a gridded ion thruster, it says that no power is consumed by the grids if nothing impacts the grids.

But I suspect that that's a situation like the way in which you can compress a gas in a cylinder without raising the temperature: "all you have to do" is to move the piston only when it will not contact a gas molecule. All the molecules will bounce off the piston while it's stationary, and the temperature won't go up. However, that requires a Maxwell's Demon to supervise the motion of the piston, and it's inevitable that the temperature goes up.

Likewise, in this situation, you simply cannot avoid having ions impact the grids, and that's where power is consumed.

However, in a normal gridded ion thruster, a lot of energy goes into ionizing the reaction mass, and you have to carry the reaction mass. This merely does away with the need for that.

I think there is no theoretical reason to assume a minimum hit rate of the charged particles. If the distance of the plates is a few meters/yards, you can have a mesh of very thin wire with holes of 1cm/.5inch. The wire can be very thin. The effect of the holes is a change in the electric field, but the deep of the region where the change is important is roughly equal to the size of the holes.

And you can build a bigger device, where the distance of the plates is 200m/200yard and the holes have 10cm/5inches and use the same wires. Or multiply the design by 100x. A much bigger device is not technologically practical, but it shows that there is a theoretical problem.

---

In their description they clearly explain that the electrons are reflected, and the protons pass though, so their design will not work. I think that if they absorb the electrons, use the protons as in a gridded ion thruster and then use the absorbed electrons to neutralize the ions, then it may work. You must use some energy to move the absorbed electrons from the front to the back of the ship because there is an electric field. This is not the device they are describing, but at least with this modification it may work.