Sorry for the way my comment is structured, it's already morning here and I did my best to try to get my point through, I haven't had the time to read the hole text yet, but I did read the 8º Chapter, and I must say that yes, the application of that mistaken dialectical method does lead to some problems when exposed to more abstract matters like society and economy.
I do note now that this mistake comes from attempting to use the dialectical method created by Hegel in the same way that the ancient Greeks used to find better answer through a dialogue between opposing parties, which is where "Thesis", "Antithesis" and "Synthesis" are used correctly.
But Hegel's Dialectics is not about two opposing sides culminating in a third one through conversation, but about the better and more developed side, in his idealistic view the big "idea", and the lesser and flawed side, the reality, struggling and ever changing attempting to reach the former.
Marx then comes and turns it on his feet, as Engels would say, and adapts Hegelian dialectics to the real and material world, consequently changing the cause of development from the contradiction between the idea and the existing, to the internal contradictions present in every existing thing.
Now what this change in method causes can be exemplified in your brothers and bikes problem: Using Diamat for them to be in there would need to be a process which puts them in opposite sides, in this case, "who is the fastest?", in this process they are in contradiction trying to overtake each other, now one brother has a better bike and consequently becomes the main aspect of the contradiction and therefore "the fastest", but given time the other brother can struggle in an attempt to improve and surpass the other brother, if he is successful in his struggle, if his struggles were enough, he can then surpass his brother as the main aspect and therefore become the fastest.
What were the differences between methods: 1º: the brothers are not contradictory as a whole with each other, but in a specific process. 2º: No dialogue is necessary, the problem and the changes come from the facts themselves(not saying dialogue doesn't help). 3º: the resolution to the contradiction do the process is in the process itself, there isn't necessarily a need for a third party to come and resolve the contradiction. 4º: Struggles are not a negative thing, but the necessary action to bring forth change. 5º: The unity of opposites means that their place in the process is interchangeable, when one becomes the faster the other becomes the slower and vice-versa.
These differences become very important when analyzing more complex situations because the entire Marxist theory is on the bases that the answer the two main contradictions of capitalism, being the private property of the means of production and the socialized production, and the organization of factories and the anarchy of production, lie in themselves, meaning the socialization of the means of production and planned production. And that those changes will only happen when the newer revolutionary class takes the place of the older reactionary class as the leaders of society.
I think I understand your point of view, your are viewing contradictions as creators of pressures that lead to a instability in the system which must be solved in a way that is in tune with both sides of the contradiction.
But what dialectical method gives us is that those pressures are not necessarily created by the contradictions, they can and, even more when dealing with nature, normally are external, what lies within the contradictory process itself is the answer to those pressures, meaning that one side must triumph over the other for this contradiction to be "solved".
So basically you are looking for the cause of the "problem" within the system but the answer outside it, when with diamat the cause is always a combination of external and internal conditions, but the solution lies within the system itself.
There is an easy example of this in thermodynamics itself, which is as much as my knowledge on the matter will allow me, that is boiling water: The water has its on internal contradiction between its liquid and gaseous state, when we as a external force apply heat to liquid water we create a rapid change in it forcing it to become, after enough heat is used, gaseous.
So in this case the cause of the change was mostly external, but the result is to be expect within the internal contradiction, as Mao would say: "In a suitable temperature, an egg becomes a chicken, but in no circumstances can a rock become a chicken".
Also I should have done this earlier, but all the points I made in these comments come from F. Engels' "Socialism Utopian and Scientific" and "Ludwig Feuerbach and the End of Classical German Philosophy", and sometimes with the help of M. Cornforth's "Materialism and the Dialectical Method".