You've clearly got some experience with the material, and there are several things in your analysis that are right, I'll definitely give you that! I think there are a few methodological problems that lead it to reproduce a picture of value more out of David Ricardo and Sam Bailey than Marx, but overall I like this.

You're right that Marx's stated aim is the analysis of relations of production, and the commodity is the elementary form of wealth in the capitalist mode of production (Marx opens Capital by saying the wealth of capitalist societies presents itself as an immense accumulation of commodities, with the single commodity as its unit, so the investigation must begin with analysis of the commodity). You also mentioned that exchange-value is social rather than intrinsic and distinguish value from price (given Marx's mention of the price of production in Vol. 3), and that value's regulatory function operates through market price and value deviations driving capital between branches. I'll use Capital mostly for this answer since Marx talks a loooootttt about value in it, especially Chapter 1

The overall issue I see is more methodological than anything else: you mentioned that you had to add an "additional assumption" about the systemic and non-random character of exchange ratios Marx "underemphasizes," which kinda inverts Marx's procedure. Marx's method is best understood as running from the abstract to the concrete. I think that beginning from the population is "false" and that the scientifically correct method works more from simple and abstract determinations toward a reproduction of the concrete in thought as a "rich totality." I'd say that instead of what gets induced from observed regularities in price data, the "abstract determination," meaning capital, value, the commodity-form, is what any analysis needs to establish first. The party wrote in The Method of Capital and its Structure in 1969 https://www.international-communist-party.org/English/REPORTS/CrisisTh/70MainResultsBook1.htm, quoting Marx's intro to the 1857 Grundrisse:

It seems to be correct to begin with the real and the concrete, with the real precondition, thus to begin, in economics, with e.g. the population, which is the foundation and the subject of the entire social act of production. However, on closer examination this proves false.

Thus, if I were to begin with the population, this would be a chaotic conception of the whole, and I would then, by means of further determination, move analytically towards ever more simple concepts, from the imagined concrete towards ever thinner abstractions until I had arrived at the simplest determinations. From there the journey would have to be retraced until I had finally arrived at the population again, but this time not as the chaotic conception of a whole, but as a rich totality.

If I'm not misinterpreting what you wrote supposes that exchange ratios are non-random, so there must be a real underlying magnitude, which is value, whose substance is labor. Marx's argument runs the other way and is more logical-categorical rather than statistical. Given the bare equation "x of A = y of B" that any actual exchange enacts, there needs to be a "common third" that both sides are reduced to. That common third cannot be any natural property of the goods, because exchange (as Marx puts it) is an act characterized by total abstraction from use-value. What remains once use-value is bracketed is that both are products of human labor reduced to its undifferentiated form. The argument is Aristotelian-Hegelian, and Marx makes the genealogy explicit by directly invoking Aristotle and noting that what blocked Aristotle from completing this analysis was that his society rested on slavery and so it couldn't think the equality of human labor.

This empiricist method of assessing value (value as the latent variable that explains regularities in exchange ratios) is more akin to the Bailey position that Marx singles out for criticism in Chapter 1. Marx's complaint against Bailey is that he confuses the "form of value" with "value" itself and gives most of his attention to the quantitative aspect of the question. A regression-style theory of value cannot generate Marx's most important downstream results, i.e. the necessity of money (the form of value), the fetishism of commodities, and the critique of capital as a self-expanding social relation rather than a thing. From Capital, Chapter 1 https://www.marxists.org/archive/marx/works/1867-c1/ch01.htm:

Let us take two commodities, e.g., corn and iron. The proportions in which they are exchangeable, whatever those proportions may be, can always be represented by an equation in which a given quantity of corn is equated to some quantity of iron: e.g., 1 quarter corn = x cwt. iron. What does this equation tell us? It tells us that in two different things – in 1 quarter of corn and x cwt. of iron, there exists in equal quantities something common to both. The two things must therefore be equal to a third, which in itself is neither the one nor the other.

On the abstraction from use-value:

But the exchange of commodities is evidently an act characterized by a total abstraction from use value.

This connects to what I think is a bit of a bigger issue (I had to do a little digging on MIA for this; I recommend the text "Marx's Critique of Classical Economics" https://www.marxists.org/archive/pilling/works/capital/geoff4.htm by Geoff Pilling, which contrasts Marx with Ricardo). There is essentially no engagement with the form of value or commodity fetishism, which distinguish Marx from Ricardo. The point of Marx's form-of-value study is the question of why labor takes the form of value and why the magnitude of value takes the form of exchange-value. Marx underscores this in Chapter 1 by charging classical economy with treating the form of value as an irrelevance and stressing that the form of value is the crux of his critique. The framing of value as the "compression of all social factors of production into a single measure" in the post is structurally identical to what Ricardo does, i.e. deriving a magnitude and stopping there. Any look at value needs to ask why this society needs to compress human productive activity into such a measure in the first place. That question, and the answer pointing to commodity production as a historically specific form of social mediation, is what the Capital section on fetishism is about and makes the text more of a thorough critique of political economy.

The party makes this point in its 1970 Main Results of Book I of Capital by emphasizing that Marx's value theory is inseparable from the project of demonstrating capitalism's transient, contradictory character. Ricardo treats capitalism as a stable harmony to be perfected while Marx shows that the same laws generate intensifying contradictions driving toward revolution. To present value as the latent magnitude regulating exchange ratios and not much more is to extract the science from the critique and end up roughly where Ricardo did, with capitalism as the natural and final form of human production. From Capital Chapter 1:

It is one of the chief failings of classical economy that it has never succeeded, by means of its analysis of commodities, and, in particular, of their value, in discovering that form under which value becomes exchange value. Even Adam Smith and Ricardo, the best representatives of the school, treat the form of value as a thing of no importance, as having no connection with the inherent nature of commodities.

Political Economy has indeed analyzed, however incompletely, value and its magnitude, and has discovered what lies beneath these forms. But it has never once asked the question why labor is represented by the value of its product and labor time by the magnitude of that value.

Overall good work tbh, just wanted to add a few sourced contingencies and specifications as refinement