The question seems a bit like: I have a half-assed quasi-formalization of a cartoon version of science, why am I getting a weird conclusion from it? Okay, that was a bit harsh (please forgive my rhetorical flourish), and the idea presented isn't totally wrong, it's just that it over simplifies things to the point where you've lost some important details.
Since you've tried to formalize the problem and discuss it in terms logical proof, your question calls to mind the ideas of Logical Empiricism.
Supposed you put on a Logical Positivist hat and tried to formalize all of the evidence as well as TENS itself, in a manner along the lines of Carnap. Could it work out that that formal version of TENS is a theorem derivable from E? Sure, it depends on what E is.
Indeed, sometimes the idea of evolution, when cast as "survival of the fittest" is viewed as a tautology, and thus necessarily true; "survival of the fittest" is not really a complete description of the theory, but being a tautology it would enter the formalism as an axiom within this project, an axiom that gets you very close to proving TENS ... maybe.
This might seem like a stretch in biology, but it happens regularly in high energy physics (and probably other places in science that have a similar degree of mathematical formalization). For a simple case, if you take conservation of energy, momentum and angular momentum as axioms, and observe neutron decay (and a bunch of axioms that allow for the interpretation of these observations), then the neutrino must exist.
In some (many?) cases this kind of formal description and analysis can be useful, if only to really clarify a representation of what you're talking about, but it can't be the end-all be-all of science for all of the reasons that Logical Positivism/Empericism folded.
I'll leave off with what I think are a few additional salient points:
Scientific proof is not universally defined as formal logical proof amongst scientists nor especially in common parlance. Different scientific communities have different methods and criteria for establishing "proof"; compare high energy physics to medical science. This almost always involves some form of empirical verification, and is usually expressed in a statistical sense, which is another deviation from formal logical proof.
TENS, when looked at in detail, is not "one theory" it is a class of ideas. E.g. I'm under the impression that it is an open question whether group selection is a real (in some sense) phenomenon (people in the field could probably pick out other open questions); which ever way this question, or the other open questions, pan out, you're still dealing with TENS.
Observations can rule out entire classes of scientific ideas. If you get enough observations rule out a bunch of other ideas, such that for everything that is left as a possibility is still in the class of TENS; then you've "proven it"; again, I've seen this more in physics (more mathematically rigorous) where experiments rule out enough different possibilities that what is left is some variation on one kind of idea.
I get the impression that the term "demonstrated as a good explanation" is a better way of understanding what "scientific proof" means. Demonstrated in the sense of having evidential support and good entailing some combination of empirical adequacy, generalizability, simplicity etc.