The biggest piece of rigor is the Austrian treatment of value, rationality, and knowledge.

Value

Austrians consider value to be subjective, or at the very least that people will act as if it is subjective. No other tradition rigorously maintains this precept. Classical and Marxist economists consider value to derive from labor, and neoclassicals see it as deriving from cost more generally.
In addition, Austrians don't consider value and preferences to be static, but that they will constantly change and shift, even moment to moment, depending on the individual.

Rationality

Neoclassical economists essentially consider all people to be purely rational economic actors. Behavioral economists question this precept, but end up maintaining that we are permanently irrational.
Austrians focus more on actions being purposeful; people take intentional actions that make sense to them given their immediate situation. In addition, they see people as capable of learning and improvement, becoming more rational over time. This is a far more realistic view of the human condition that produces more rigorous results.

Knowledge

The neoclassical school considers knowledge to propagate instantly throughout society. Austrians such as Hayek reject this. Contemporary Austrians have actually started to study how quickly knowledge will propagate, which is a far more rigorous framework and research project.
I fundamentally disagree with everything you said about value. At no point in my several years of economics training did I learn that value is derived from labor. Quite the opposite actually. The only time I heard that value was derived from labor was in philosophy of economics when covering Marx.
I also disagree with your premise "no other tradition maintains this precept". That is only because the vast majority of economists don't consider themselves part of a tradition. They just do economics research. There is a whole host of people who do research that would align with various Austrian viewpoints.
Please define how you are measuring the rigor of a result. Nevertheless, suppose I grant that this is true, you are assuming that the rigor of a result is the metric to prioritize. What is the value of a rigorous result without understanding e.g., precision?
Let me suppose the rigor of a result is measured by the generalizability of results. IE, if minimum wage is above market wages, unemployment will increase. This is a rigorous result under my supposed definition as it is widely generalizable. Why is that better to know than something less generalizable but more precise?
As far as knowledge goes, Non-Austrian economists call this friction. Models assume no friction to glean insights. All economists know that in the real world there is friction.
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Classical and Marxist economists consider value to derive from labor,
Really? Who in the modern era, other than Marxists, buys into the labor theory of value?
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Basically, no one: The Classical economists were Smith, Ricardo, Mill, Malthus, and Marx.
Most modern economists are neo-classical, which is sort of classical + subjective value theory.
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Nice summary. Value and Rationality represent core Austrian insights that the rest of the school of thought is built on. The propagation of knowledge is one of the main theoretical contributions. At least that's how I look at it.
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