In order to understand why the direct band gap at K in the primitive cell is not at the K point for the supercell you have to understand Bloch's theorem(

https://en.wikipedia.org/wiki/Bloch%27s_theorem) and what the Brillouin zone (

https://en.wikipedia.org/wiki/Brillouin_zone) actually is. The Brillouin zone for a supercell is not the same as for the primitive cell.

Let's consider a simple example in 1D. The 1D unitcell has length L. Then the k-points are given by k = t 2π/L, where t is the "fractional k-point", i.e. the coordinates inside the first Brillouin zone. The first Brillouin zone is defined as the coordinates of t for which -0.5 < t ≤ 0.5. If you have a fractional k-point outside this region it is wrapped back inside. Now if you consider an equivalent system, but where instead of using the primitive cell you consider a supercell of length 2L, then the k-points are given by k

_{2} = t

_{2} 2π/(2L). Likewise, the Brillouin zone of this system is defined as -0.5 < t

_{2} ≤ 0.5. You'll see that in terms of k, the Brillouin zone of the supercell is half the size of the primitive cell. Now the k-points of the states in the two are still the same. Let's consider the states at the K point of the Brillouin zone of the primitive cell. The K point is determined by t

_{1} = 0.5. We have k

_{1} = k

_{2}, so 0.5 * 2π/L = k

_{2} = t

_{2} 2π/(2L) => t

_{2} = 0.5 * 2 = 1.0. So the K-point of the primitive cell corresponds to the t=1.0 fractional k-point of the supercell - this gets wrapped back to the Gamma point (t

_{2} = 0) in the first Brillouin zone of the supercell.

If you want to consider a band structure of a supercell as projected on the primitive cell you can use the effective band structure analysis tool (

https://docs.quantumatk.com/tutorials/effective_band_structure/effective_band_structure.html).

Also, if any of this sound confusing I suggest you to revisit your basic solid state physics book and do some example calculations by hand.