Even if Marx's thinking changed in between Vol 1 and Vol 3, that's not a good thing for the theory and cherry-picking is still cherry-picking.

! The point isn't that Marx changed his mind, but that, among other communication problems, he makes many broad statements and then carves away caveats, either afterwards or in some definition elsewhere (that hopefully you've remembered correctly). For this particular example, though, consider the paragraph that you cut out with an elipsis:

Some people might think that if the value of a commodity is determined by the quantity of labour spent on it, the more idle and unskilful the labourer, the more valuable would his commodity be, because more time would be required in its production. The labour, however, that forms the substance of value, is homogeneous human labour, expenditure of one uniform labour-power. The total labour-power of society, which is embodied in the sum total of the values of all commodities produced by that society, counts here as one homogeneous mass of human labour-power, composed though it be of innumerable individual units. Each of these units is the same as any other, so far as it has the character of the average labour-power of society, and takes effect as such; that is, so far as it requires for producing a commodity, no more time that is needed on an average, no more than is socially necessary. The labour-time socially necessary is that required to produce an article under the normal conditions of production, and with the average degree of skill and intensity prevalent at the time.

This is an explicit disavowal of the simplest possible interpretation of the labor theory of value, the claim that "labor in = value out." The 'labor' he's talking about isn't actual labor, but hypothetical generalized labor, and it basically exists just to be a natural currency.

let me just point out that the labour theory of value is widely accepted by Marxists (those that still remain) as the correct one, see e.g. here.

Sure, and I agreed on that point up here. LTV, as I see it, runs into huge Motte and Bailey problems, where the actual theory is inoffensive as a "dollar theory of price," in which the 'price' of a commodity is the 'number of dollars it costs', but the conclusions people want to use it for are "the past doesn't matter, everyone owns everything!" As far as I can tell, though, Marx stayed in the motte, and this is where the "I'm not a Marxist" statement comes from.