The problem is that, given the final expression 6 ÷ ½ * 3 we don't really know where it came from. OF COURSE if we know is the result of calculating the area of a triangle we could maybe infer that ½ * 3 is half the length of a side or a height and then the only sensible thing to do is group that. But triangle areas are not the only subject we could arrive this expression from.

For example, let's take another super-basic mathematical calculation, the cross-multiplication, where you know that a couple of ratios are equal: a ÷ b = c ÷ d, and you must calculate one of the topmost variables, either a or c (direct cross-multiplication). For example:

a ÷ b * d = c
c ÷ d * b = a

Now we should conclude that a ÷ b and c ÷ d are one term, even without parentheses, right?

So that:
1 ÷ 2 * 6 = 3.
3 ÷ 6 * 2 = 1.

And therefore we would consider these other calculations (that use your logic of grouping the multiplication) nonsensical:

The problem is that, given the final expression 6 ÷ ½ * 3 we don't really know where it came from. OF COURSE if we know is the result of calculating the area of a triangle we could maybe infer that ½ * 3 is half the length of a side or a height and then the only sensible thing to do is group that. But triangle areas are not the only subject we could arrive this expression from.

For example, let's take another super-basic mathematical calculation, the cross-multiplication, where you know that a couple of ratios are equal: a ÷ b = c ÷ d, and you must calculate one of the topmost variables, either a or c (direct cross-multiplication). For example:

a ÷ b * d = c

c ÷ d * b = a

Now we should conclude that a ÷ b and c ÷ d are one term, even without parentheses, right?

So that:

1 ÷ 2 * 6 = 3.

3 ÷ 6 * 2 = 1.

And therefore we would consider these other calculations (that use your logic of grouping the multiplication) nonsensical:

1 ÷ 2 * 6 = 1 ÷ 12.

3 ÷ 6 * 2 = 3 ÷ 12 = 1 ÷ 6.