What you might be missing is that the shear failure section is at a 45 degree angle. For a crack to form at the face of the support at the bottom of a slab, the top end of the crack is a distance away from the support equal to d.

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Why yes, I do in fact have no idea what I'm talking about

I understand that the crack formed would be at an angle. But, in which cases do we take the critical section to be at the face of the support? Also, wouldn’t it be more conservative to always take the higher shear value produced at the support location (for simply supported and cantilever beams)?

It is more conservative to take the shear at the support and also makes calculation much simpler, so it is not uncommon/bad to use the shear at the support as the design value. Generally speaking for concrete there is no requirement to do this however. If the crack was any closer to the support (i.e. the crack starts passing through the column) the column itself will resist the shear/vertical forces and the section is not critical. I've probably worded this wrong in a technical sense if someone could correct me

Thinking about it in very abstract terms, the slab isn't going to fall through the column, and the column is going to be able to take the vertical load by design.

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Why yes, I do in fact have no idea what I'm talking about