The theory is that surplus value (which can become profit, or be reinvested, etc) exists only through paying the worker less (on aggregate) than the aggregate amount that the worker provides through their labour.

The mainstream economics answer to this is to say that the labour theory of value is bunk and that prices are defined purely by market forces. But this kind of misses the bigger picture of what Marx was getting at rather by getting lost in the weeds about price definitions and really is a critique of Ricardo more than Marx.

Despite using the term "exploitation" which has moral overtones in English, Marx really isn't casting a moral judgement here. It's a technical description. But he does believe that this phenomenon leads to structural inequality and injustice and that the only resolution is some other kind of property ownership or method of production. Though he was famously vague on what that alternative would be, and people far less intelligent than him were left to fill in the blanks with some rather awful alternatives.

This is very true, and cannot be overstated. Often, the criticisms of one theory are actually much better direceted at a predecessor theory. Students in a history of economics course, or a regular economics course, may learn of the "labour theory of value", but they may learn either Smith's, Ricardo's, or Marx's, and assume that the theories are the same, or that the same criticisms apply to all of them.

There are good and serious criticisms of the "labour theory of value" from those unsympathetic (and even those sympathetic) to its apparent normative conclusions. It's a shame that many people seem to confuse one thinker for another, especially when each boasted of their improvements (Marx, for example, writes "I was the first to point out and examine this [...]" when writing on the "LTV" and discussed the law of the tendency for the rate of profit to fall again in his original understanding, despite being extremely well read on - and critical of - his predecessors).

Crucially the value wasn't created by the worker, but rather the worker played a part in the creation of that value. In a capitalist society, that worker's wage is determined based on the value of their contribution and the supply of workers with the requisite skill. In a Marxist society, it is set by the state, and it seems to be very hard for states to determine a price that is sustainable.

In my humble opinion, Marxism seems like a denial of basic economic realities--namely that the state can set the price of anything to whichever value they prefer and it won't have disastrous economic ramifications ("price is just an arbitrary number"). Maybe I'm creating a straw man, but I do notice a lot of Marxists who make arguments about what is fair and not what is economically sustainable as though economic sustainability is an invented problem that we can disregard. Perhaps this view isn't uniformly shared among Marxists, in which case my criticism is "too many Marxists are making these kinds of unconvincing arguments, and more convincing arguments would be centered around economic feasibility--we all agree that we want poor people to have more money, the question is 'how?'".

I think we need to define "value of their contribution".

If a friend of mine has an idea for an app, but he has zero programming knowledge and so he pays me $1,000 to develop it, and he ends up making $1,000,000 from selling it, what is the value of my contribution?

Is it only $1,000 because that was the agreed price for my labor? Or is it $1,000,000 because that's how much money he made from it? Or something in between?

Obviously, in this hypothetical, I should be negotiating a share of the revenue instead of just a flat fee, but generally most workers don't get to do this.

So...what was the value of my contribution?

If your friend insists on paying you a low flat fee for doing highly lucrative work, your rational choice is to turn down the offer and find another employer, or even go at it on your own. Your friend's rational choice is to negotiate because the lost profits from you (and others) declining the offer is worth more than the profits he loses by paying you more.

Regardless of that, a single transaction like this is often not sufficient to accurately determine the market value of your contribution. Market valuation happens in aggregate for this reason. A single irrational transaction like the one you described just represents a market inefficiency.

He did feel that some other model of ownership & management could lead to some other form of remuneration, but he definitely left it unclear.

Read the other discussions on this thread, there's far more nuance here than I think you're understanding.

If a Marxist is one who says "Capitalism is imperfect; I've got a gut feeling that there is a better system out there, but no idea what it looks like concretely", then that may be correct but it's close to worthless.

If a Marxist is someone who says "Capitalism is imperfect; the state needs to fix the price of $X (labor, etc)" with no mention of the economic feasibility of said price-fixing, then that's better but still far from persuasive.

My criticism was directed at the latter. There are perhaps many good Marxists who make convincing economic arguments; I'm asking for more of this.