First: As (somewhat confusingly, IMO) stated in the other answer, the use of "brachistochrone" is somewhat of a misnomer, potentially. I will use the term "constant-acceleration minimum-time transfer".
Second: "Efficiency", in spaceflight, is a somewhat complicated and many-sided thing. Efficiency of cost? Total spacecraft mass lifted into orbit? Amount of energy? Duration? Fuel?
Simply to travel along that blue line, which makes a sharp angle with the orbit of both Earth and Mars, and which travels backwards relative to the orbits of the planets (on the trip from Earth to Mars) requires an immense amount of propellant compared to a more normal trajectory like the one on the original drawing. Having an ion drive doesn't change that fact (though it does make the amount of delta-V somewhat more achievable. Keep in mind that ion drives are so low in thrust that they may take months just to escape from the orbit of Earth -- in which time Earth and Mars will both have moved a significant distance in their orbits.) A constant-acceleration minimum-time transfer along that line is even worse -- taking thousands of times more delta-V than even just a basic trajectory along that line.
If time is of the essence, and you have access to technology that is currently only science fiction (imagine an engine that is as powerful as exploding several thermonuclear warheads a second... inside the ship's rocket nozzle) then this is practical. Otherwise, the need to save propellant, fuel, or money building the engine will make more limited options more efficient.