You probably have a small box [...] People who told that story were, of course, well aware that [...] snakes don't talk

I think you've missed a central point of my argument. (So I wasn't clear enough. Sorry.) Of course they were well aware that ordinary snakes these days don't talk. But reading the story, it seems to me clear that they thought it perfectly reasonable that once upon a time the smartest non-human animals might have talked, and that snakes might have been the smartest non-human animals.

In other words, it seems to me that this is not a story about a snake given special powers by magic, or sufficiently advanced technology, or devil-possession. And that's exactly why the fact that the snake talks indicates a deficiency in their understanding. If the story had said "Now, an evil spirit had entered into the serpent, and it spoke: ..." then it wouldn't have had been evidence of that deficiency.

the conventional Christian interpretation

Christian theology isn't homogeneous enough for there to be such a thing as the conventional Christian interpretation, but here are some comments from a few different sorts of fairly-mainstream Christian sources.

The NIV Study Bible (conservative evangelical, fairly lowbrow, creationist) takes basically the position you describe. "The great deceiver clothed himself as a serpent, one of God's good creatures".

The New Jerome Biblical Commentary (liberalish Catholic, middle-high-brow) takes a precisely opposite view. "The snake is not Satan, though later traditions so interpreted it [...]. He is simply a mischievous creature made by God, dramatically necessary [...] he recedes into the background when his narrative function is accomplished."

"The theology of the Book of Genesis" by R W L Moberley (liberal Anglican, accessible academic) concurs. "So, although the serpent is clearly not identified with Satan, as in much subsequent construal of the story, the implied reader has good reason to be wary about words from an archetypal ancestor of enmity -- indeed, potentially deadly enmity -- against humanity." (He is referring here to nothing more theological than the fact that snakes and humans don't tend to get on.)