speaking, rigidty increases the more the flat section rotates from horizontal to vertical. In practice, though, today flat-section bars are either postitioned horizontally (minimum rigidity) or vertically (maximum rigidity).
Finally, “additional” bars haven’t always been removable. In the past, some karts had the front bar welded to the frame or cut in the middle. In the first case, you couldn’t adjust stiffness, while in the second you did so by central lock-down of the two sections. Bars used to be fixed to compensate for the structural weakness of the joints between bar and chassis, which causes non-uniform flexing overall. Today, manufacturing technicques are so sophisticated as to make this issue non-influential. In fact, almost all chassis on the market come with a removable front bar.

Coming back to chassis bars, we yet have to speak of the lateral kind. This goes on the left side of the frame, at seat height, and adds stiffness to the central section of the chassis, causing an overall increase of frame rigidity, rather than having a localized effect on front or rear end torsion. This means that all four tires gain grip, but also that the kart becomes more “edgy”.
Now, a counter-inutitive “paradox”: an a wet track you want to remove all bars, to make the kart softer and so more “malleable” in direction changes. This helps you remain in control of the vehicle, especially coming in and out of corners (due to prolonged understeering in the first case and oversteering in the other in low grip conditions). The reason why adding stiffening bars in wet conditions won’t produce the desired effect of greater grip is that traction is just too low in absolute terms.