In meiosis chromosomes segregate on the basis of random assortment. So in theory it is possible for an organism to produce a zygote only having genetic material from one of its parents.
This is worded a bit strange. A zygote can not have chromosomes from just one parent, otherwise it would still be a haploid cell. I think you meant to say that each parent could pass on chromosomes from just one grandparent on that side. For example, the mother could pass on chromosomes that came just from her father and none from her mother. Or you could replace "zygote" with "gamete" in your original post and it would make more sense.
For 23 pairs of chromosomes the chances of this occurring are 1 in 2^23 or about 1 in 8 million. It is the same odds of flipping tails 23 times in a row.
If you accept this to be true, how do you fit Mitochondrial Eve into the scenario?
How does it not fit into this scenario? For every gene there will be a most recent common ancestor (MRCA), and they will undoubtedly be different MRCA's for many genes. Our genomes are undoubtedly made up of DNA that was not found in ME but was found her contemporaries.
Let's use language as an example. There are many different words in the english language. It would be incorrect to trace the origin of a single english word and date the origin of the entire language based on the origin of that single word. And yet, this is exactly what you are doing with ME. Our mitochondrial genome is but a single piece of our entire genome. There are many different points of origin for different genes just as there are many different points of origin for different words in the english language.
If all human beings alive today have a particular gene (marker) and if this can be traced back to a point where it originated, it is only obvious that we all descended from that individual. No?
Every gene in the human population could be traced back to a most recent common ancestor. The point is that different genes will have different MRCAs. We can even apply this concept to different species. For example, ERV's shared by humans and chimps came from a common ancestor.