Why does shifting under power affect shift quality? I recently was attempting to shift up a hill and observed that 


The shifting levers on the handlebars displayed significant resistance to my pushing them.
Despite several rotations of the crank, the bicycle had not shifted gears.


Googling around, I've seen people claim that pedaling under power increases chain tension and that this chain tension will affect how easily the chain moves between gears.

Intuitively, this does not make sense to me, as how easily the chain moves orthogonal to the direction of travel (i.e., the ease of shifting) would seem to be independent of the chain tensions parallel to the direction of travel. This seems to imply there's a causal relationship between the relative magnitudes of these forces.

Moreover, given that the chain itself is the same length regardless of the power transmitted, it's unclear to me how pedaling increases chain tension. Is it that one part of the chain becomes more tense, and the other becomes less tense? A free-body diagram type explanation would be very helpful.

How might one understand or model the shifting action and mechanism in a way that explains why chain tension affects shifting?
There may be other factors that affect "shift quality", if I understand what you meant correctly, but the most significant by far is “direction of force applied.

Basically for a gear to change the mechanism just pushes the chain “off rail” enough in the direction of the new gear. When the chain has a higher tension applied on it the derailleur doesn’t have enough strength to push the chain “off the rail”. (Derailleur is the name of the part the pushes the chain)
If you still haven’t understood, imagine a line stretched between two fixed poles, if it is loose and you push it to the side it will move some, if it has a higher tension and you push to the side with the same strength and direction than before it will move less. And the tighter the line the less to the side it would move. A gear system has a minimum distance that it needs to move to the side so it can change gears.

More advanced, modern and obviously expensive have over time placed gears closer to each other and shaped individuals to make the “derailing” need a smaller push in a new direction, requiring less strength to push the chain and as a whole make the process more smooth.

If you didn’t understand something just ask.