The thing is a large number of the public defender Darwin's ToE also do not demonstrating the most basic understanding Darwin's ToE .... Darwin's ToE does not explain (nor try to) how life arose, but the general public I would say generally does think that is does...
Darwin's ToE describes a mechanism how a living thing that replicates itself (with and from it' slight variation) can and will create diverging forms of life.... not the creation of life itself (from non life).... there are some theory that do try to explain the creation of life via a ToE like mechanism....but does Darwin's ToE?
The short answer is "no" -- Darwin's seminal work was entitled The Origin of Species, and not The Origin of Life, because he has nothing to say on the original beginning of life itself (abiogenesis).
What Darwin did write, at the conclusion of his book,is, I think, very well put:
There is grandeur in this view of life, with its several powers, having been originally breathed by the Creator into a few forms or into one; and that, whilst this planet has gone circling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being evolved. [My emphasis]
Your point, that "the general public" think Darwin theorised about the origin of life itself is true, but it is wrong. And it is an error, I think, which is promulgated by "anti-Evolutionists", whom I suspect of having a specific, and very narrow, religious sectarian view, and a political agenda to match.