Ahsan Allahu Ilaykum Ustadh Al-Karim, we are still confused regarding what's a pillar of ṣalāh. Is the correct definition that which is during the ṣalāh, is continuous throughout the ṣalāh until the end of salah, and sujud as sahw does not mend it if left out due to forgetfulness or ignorance?

The rukn is a part of the salah itself, and its mark is that it does not drop by forgetfulness or ignorance: it must be performed, and sujud as sahw does not stand in its place, unlike the wajib, which drops by forgetfulness and is mended by that sujud. Continuity through the salah is nearer to the description of a shart, like purity and covering the awrah, which stay outside the reality of the salah rather than being a part of it. So, it starts before the salah until it ends. As for the pillars, their continuity is that they have to be fulfilled until the worship finishes.