Cookies – Why do Pocky and other “stick” biscuits have this distinct burn pattern

This looks to me like the symptoms of too much flour - probably way too much.

Measuring baking ingredients by volume instead of weight is always a crapshoot, although it shouldn't be that far off - my guess is that either the flour was heavily compacted, or perhaps the baker's mind wandered off and she accidentally added an extra cup.

Overmixing (i.e. by using an electric mixer) is a common problem with cookies but the symptom of that is a hard and gritty cookie that doesn't rise - not one that rises too much and has unbaked flour in the center. As far as I know, that can only happen with a huge amount of flour. Obviously, too much flour would also result in a much denser dough.

What you ended up with seems to be more like a quickbread - almost a scone - and there is indeed not an awful lot of difference between the two aside from the ratios and the creaming of butter.

Excess flour probably wouldn't result in a stickier dough, but stickiness is more due to temperature than anything else. If you're finding the dough too sticky to mold then make sure it's thoroughly chilled - that will make it easier to work with.

Mixing it with milk (or liquid in general) is probably just to remove clumps. It clumps easily, especially when things aren't perfectly airtight (probably more common in your grandmother's time), and mixing in a small amount of liquid is an easy way to make sure it all breaks up.

I'm less sure about the alternating. It is pretty common to alternate wet and dry as a way of getting things evenly mixed and avoiding lumps, so it's possible it's a variation on that. The paste might be too thick to easily mix into the wet, but also wet enough that it'd tend to stay as a lump if you mix it straight into the dry, so alternating wet-dry-paste might get you more even mixing? But if you're able to mix the paste smoothly into the wet, it's completely fine to just do that.