I only have a layperson's knowledge of evolutionary biology, so my criticisms might miss some important subtlety, but it seems to that your analogy is significantly misleading in a couple of ways. It does convey the idea that random guesses with incremental feedback is a better search strategy than if the feedback were holistic (e.g. if you were guessing whole words and the only feedback were whether the guess is correct or not). In so far as someone's worry about natural selection is that they're mistaking it for the latter sort of search, the analogy may be helpful. But if you want to convey something more specific about how natural selection works, then I'm afraid the analogy isn't all that great.

One drawback of the analogy is in the nature of the environmental feedback. In Hangman, a letter gets fixed if (and only if) it is part of the correct answer. In genuine natural selection, though, a mutation doesn't get fixed because it is part of a complex set of mutations that collectively confer some phenotypic benefit. The environment isn't forward-looking like that; it doesn't say "This mutation is part of what is needed for optimality, so I'm going to hold onto it for that reason." Each individual mutation, in order to get fixed in the population, must confer some immediate reproductive benefit. Merely being one element of some complex group of mutations that is collectively beneficial is insufficient. The hangman analogy doesn't capture this aspect of natural selection.

This actually leads the analogy to kind of play into the hands of "irreducible complexity" critiques of natural selection. The proponents of such critiques presume that the individual parts of some complex adaptation only benefit the organism to the extent that they are part of that complex adaptation, and hence one cannot explain their selection without supposing that there is some forward-looking element to selection which holds onto those individual changes just because they will eventually contribute to a complex adaptation. This forward-looking aspect is then offered as evidence of intelligent design.

Another big drawback is that the analogy doesn't capture the competitive nature of natural selection. Natural selection occurs in populations, and requires both variation in traits among individuals in the population and competition for resources among those individuals. The Hangman analogy suggests that the environment already has a fixed template for the ideal phenotype and that it punishes organisms (or genes) individually for failing to approach this ideal and rewards them for getting closer to the ideal. If you have a population, and things worked in the Hangman way, there would be no correlation between rewards and punishments. But that's not how natural selection works. Genes are rewarded for contributing to their vehicles (organisms) being more reproductively successful than other organisms in the population. A reward just consists in reproducing more than your competitors, and a punishment just consists in reproducing less, so rewards and punishments are correlated. One allele can't get rewarded without another one getting punished.