Proper time between two events in a timelike separation

Here is what i read in classical field theory by Landau, i quote:

The time interval read by a clock is equal to the integral

$frac{1}{c}int_{a}^{b}ds$

taken along the world line of the clock. If the clock is at rest then its world line is clearly a line parallel to the $t$ axis; if the clock carries out a nonuniform motion in a closed path and returns to its starting point, then its world line will be a curve passing through the two points, on the straight world line of a clock at rest, corresponding to the beginning and end of the motion. On the other hand, we saw that the clock at rest always indicates a greater time interval than the moving one. Thus we arrive at the result that the integral

$int_{a}^{b}ds$

taken between a given pair of world points, has its maximum value if it is taken along the straight world line joining these two points.

No, your approach doesn't make sense. Given A is at the origin and B is on the $t$ axis, the straight-line distance between them is $Δτ = Δt$. There's no gamma factor. If you computed straight-line distance in an arbitrary reference frame then you would have $Δτ = Δt'/γ$. But this is simply because $Δt' = γΔt$. You're just computing the same thing (straight-line distance) in different coordinates.

To show that the straight-line distance is maximal over all paths, you need to integrate over all paths. Define a path as a differentiable function $mathbf x(t)$ with $|mathbf x'(t)| < c$. Then calculate the length of that path using the spacetime metric, and show that it's maximized when $mathbf x'(t)$ is constant. You can use any coordinate system since they're equivalent, but it'll be easiest to use the coordinates where A is at the origin and B is on the $t$ axis (in which $mathbf x'(t)$ is identically zero if it's constant).