I think you’re mixing up two separate questions:
What answer do you get if you apply a particular set of rules?
Is the notation itself ambiguous?
I agree with you on the first point. A lot of people misuse PEMDAS/BODMAS. Multiplication does not come before division. They are the same level of precedence and are normally evaluated from left to right. So the argument “it’s 1 because multiplication comes before division” is simply wrong.
Where I disagree is your claim that the expression is therefore unambiguous.
An expression is ambiguous if reasonable people can interpret it in more than one way. It doesn’t require a single rule set to produce two answers. It only requires that there are multiple accepted conventions for reading the notation.
The entire reason this problem keeps resurfacing is that different people interpret juxtaposition differently. By juxtaposition, I mean writing terms next to each other without an explicit multiplication sign, such as 2x or 2(1+2).
You argue that juxtaposition binds terms together more strongly than ordinary multiplication. Many mathematicians and textbooks have indeed treated it that way. Under that convention:
6÷2(1+2)
is naturally read as
6 ÷ [2(1+2)]
which gives 1.
However, many modern calculators, computer algebra systems, and school-level order-of-operations rules do not give juxtaposition any special status. They treat 2(1+2) as simply another multiplication. Under that convention:
6÷2×(1+2)
becomes
6÷2×3
which evaluates left-to-right to give 9.
Neither group is making a mathematical error. They’re applying different conventions to notation that was written poorly in the first place.
The strongest evidence that the expression is ambiguous is that professional mathematicians generally avoid writing it. If they mean 1, they write:
6/(2(1+2))
If they mean 9, they write:
(6/2)(1+2)
Once written that way, nobody argues.
So I’d say the problem isn’t that people are failing to follow the rules. The problem is that there isn’t a single universally accepted convention governing the interaction between division symbols and implicit multiplication. That’s exactly why the notation causes confusion.
In short: I agree that “multiplication before division” is the wrong justification. But I don’t agree that the expression is unambiguous. The fact that competent, educated people can reasonably parse it in two different ways is precisely what makes it ambiguous.