Don, I think you need to do two things here:
First, the notion that one will find a problem in evolutionary theory purely through reasoning, e.g., by identifying “fallacies,” is just incorrect. This is not a logical, but an empirical, problem, and the only way you can access it meaningfully is by engaging with specific evidence and demonstrating what you think the problem is. Others have pointed out that the thing you judge to be a fallacy clearly isn’t; but even if it were I can’t see how this would help you in the face of the evidence.
I would strongly suggest that you take up the project of better understanding some major evolutionary transition, so that you get some idea what this actually looks like. Two books I would recommend are Kemp’s Origin and Evolution of Mammals and Clack’s Gaining Ground. The first is, of course, about the origins and evolution of the mammals, beginning with the earliest, very reptile-like synapsids from the time just after synapsids and sauropsids diverged. It addresses, at length, the many structural changes and the stages through which they’ve passed. The second is about the origin of the tetrapods by way of their lobe-finned fish ancestors.
Both of these books are a bit technical for the layman. But as a layman I can also attest that if you read them carefully and diligently, you will learn a great deal; they are far from inaccessible but will require effort.
That effort, however, will be richly rewarded. Rather than having a sterile “reasoning without evidence” approach to the question, you will gain an evidence-based understanding of some very understandable examples of how complex suites of characters arise in a population without any teleology, without any need to look forward to some “goal,” at all. If, after that, you still have some critique of evolutionary theory, it will now be a better-informed critique, at the least.
P.S., added via edit:
I also have to say that it appears that you have a very broad conception of the fallacy of composition. From the way you speak of it, I would expect that you think that when you look out and find six feet of snow on the ground, this cannot have resulted from the deposit of many individual snowflakes. Again, the concept of “fallacy” is pretty useless in this context anyhow; the question is whether the evidence supports the proposition that changes in populations can accumulate, or not. But that’s pretty much just as clear a proposition as the one that says snowflakes can accumulate.