Why are these so frequently used in modern usage as to be unquestioned? I see these constructions daily in well-respected publicatiions and wonder if they are as jarring to anyone else as they are to me. Doesn’t the use of “yet” include the sense of the “and” or the “but”? This has been bothering me for twenty years and since it seems to have become more frequently used it bothers me even more. I thought I’d ask people here who might let me know if I am misremembering a simple rule. If so, this rule has its roots in logic - unless I’m doubly wrong.

I have searched old threads and cannot find the topic. Every dictionary I consult teams “yet” with either “and” or “but”.

In trying to construct a phrase in which these extra words are necessary for meaning, I fail. Why do we need an extra word in each of these sentences?
He spent hours writing the code, but yet the progam failed.
She hoped her employer wouldn’t fire her, and yet the next day she was unemployed.

Other than being my own bad writing, these sentences would convey the same meanings without an extra word, wouldn’t they? Maybe it’s just some half-remembered lesson from school and causes the same unpleasant frisson as when I see “alright” in print.

In the second example ("and yet"), the “yet” emphasizes the contrast between what she hoped and what actually happened.  “And yet” is essentially equivalent to “but” in this respect, introducing a clause that is contrary to the preceding clause. 

(Edit: I belatedly realized that your point is that “yet” alone, without “and”, would do as well.)

I agree that it is periphrastic in the first example ("but yet"), but I don’t have any sense of hearing this often, or that either construction is increasing in frequency over time.

And yet and but yet have a long pedigree in the language. A few examples from Will:

Now I’ll stand to it, the pancakes were naught and the mustard
was good, and yet was not the knight forsworn.  As You Like It, I,ii,198

My youngest boy, and yet my eldest care,
At eighteen years became inquisitive Comedy of Errors, I. i, 126

Sir, it is
A charge too heavy for my strength, but yet
We’ll strive to bear it for your worthy sake
To the extreme edge of hazard.  All’s Well That Ends Well III, iii, 1546

I’d say the cnstructs have always been very common in English. I have no problem with them at all.

Yes, both are so old and so common that usage manuals don’t even mention them, although the OED notes ”and or but may precede yet.”

It is redundant, but redundancy is not always bad. In poetry, there may be metrical reasons for using both words, and in prose the duplication adds a degree of emphasis to the disparity, calling greater attention to it.

It doesn’t seem redundant to me. “And yet” means “and, despite this”: “but yet” means “but, despite this”. It definitely adds to the meaning.

Language is not about logic, and many idioms make no sense when analyzed logically.

True but I think this one does.

Doesn’t matter.  The whole approach is misguided.

Fine…

It is redundant, but redundancy is not always bad.

Agreed. It’s just what we say.  Correction - what I say.  Which makes both okay by me.  I probably use “and yet” to agree with the preceding statement, and “but yet” to contrast the previous statement with my own opinion, about to follow*.

*Which it doubtless will, either in this thread or another.

I think this is one of many pairs of words or phrases which many will agree have some shade of difference in meaning but few will agree on what those differences are.

In which case there is no difference in meaning. Pretty much every word or phrase that has a close synonym is used idiolectically in a distinctive sense. While you are free to make such distinctions in your own speech and writing, you would be wrong to try making others do so or to rely on others to divine that you are making a distinction.