Suppose we took all living humans and found the set of their mothers, mothers' mothers, etc. and then traced down as far as possible. Is there a logical reason that this tree has to converge to one 'mitochondrial eve'?
Yes; she wouldn't necessarily have to be of the same species, but "mitochondrial Eve" must exist. The proof is pretty simple if you assume no one has more than one mother and that some mothers share a mother (and some other reasonable biological assumptions like the finite number of offspring).
Consider all living maternal lineages at any slice of time. This would effectively be a slice of all mothers, let's call it Generation N. In turn, these mothers all have mothers. We can call this Generation N-1. Not all women who have children in this generation are included: N-1 only contains mothers of the Generation N mothers. Generation N-1 must then be smaller than Generation N, because no mother has more than one mother. In practice, it's likely much smaller, since any mother who has more than one daughter who herself becomes a mother reduces the total possible by 1 more. You can recursively go through maternal generations N-2, N-3, etc this way and will find that every successive generation gets smaller. Eventually it will converge to 1 person, and that's your "Eve".
You can think about it similarly in the opposite direction, and find that as you go forward, extant maternal lineages can disappear but once a lineage disappears it cannot come back.
See also the Galton-Watson process.
A lot of "mothers" the same age as mitochondrial Eve have descendants today (in fact, they are likely to be either ancestors of everyone or ancestors of no one), but those lineages did not pass maternally. See also https://en.wikipedia.org/wiki/Pedigree_collapse