What would a transitional fossil look like?

Can I ask how do you think the genome creates the structures of the body? You talk about mutations in terms of overall damaging genes, which suggests you are thinking of genes that code for proteins. But this is only part of the story, because there is also the mechanisms which control when these genes are expressed and how they interact.
An embryo is a population of cells and as they multiply gene expression changes. So cells in a certain region form a limb bud, elongates to form an arm and finally a hand, all while the timing of when different genes are expressed changes. What you describe as differences between humans and chimps is simply differences in timing of when genes are expressed. It may not even need mutations the protein coding genes for these differences to occur.
Another example for humans and chimps is the skull development of the foetus. For both species development is almost identical, with expression of genes at first developing the size of the cranium, then there is a point when this expression is reduced and instead the expression switches more to development of the jaws. In humans this switch occurs later in development than in chimps. Edited by Meddle, : No reason given.

But that is the point, there is nothing in the genome to define what the final body structure will be. What we perceive as a final body structure is the culmination of many generations of embryonic cells multiplying and their gene expression changing in response to the cells that surround them and the signalling proteins they're exposed to.
This also means that changes earlier on in development carry a greater risk of damaging the individual as they can impact how cells interact later on in development. Therefore the innovations to allow species to diversify usually occur towards the end of development, and are constrained by natural selection, which results in your observation that body structures in related species follow a similar pattern. So dogs are dog-like and apes are ape-like.
But that also means the differences between humans and chimps are minor because they occur towards the end of development, so my example of the skull, where the timing of gene expression changes the ratio of cranium to jaw size; or your example of limbs, where gene expression determines how far a limb bud is extended into a leg and eventually the length of toes. After all, this is not much different than what differentiates a gorilla from a gibbon.