Why would an alternator fail after such low usage?

16 years old.  lack of use is worse on all car parts than mileage is

"Failed across the board" is about as unspecific and untechnical of a slang statement that you could make about an alternator. 

Alternators can fail in several ways. 

The most common way is for the carbon/graphite (whatever they are) brushes to wear out.  They are slowly consumed as the alternator rotor turns.  So if your engine is running, your brushes are wearing minutely.  This is normal wearing out. 

I have for decades, opened my alternator and changed the brushes.  It takes a few minutes and they cost perhaps 25¢ each (there are 2).  Decades ago they were like 5¢ each. 

Years ago you could get the alternator off the car in a few minutes.  It took another 5-10 minutes to open the alternator and change the brushes, then a few more minutes to reinstall it. 

Even a slow person could do the job in half an hour. 

Nowadays, alternators are more time consuming to remove/replace and a bit more difficult to crack open. 

I've been a mostly a Chrysler, Ford, GM mechanic for many years, and they've been fairly easy.  I've had a Ford Ranger for the past decade and I can get the brushes out by removing the harness connector and removing the voltage regulator from the back, without even having to remove the alternator from the engine.  4 screws hold it on.  The regulator pulls right out and is the brush holder.  It takes a few minutes to pop off the old brushes and install 2 new brushes.  Then put it back on and the job is done.  I've done it twice in the last decade.  The slip rings on the rotor (that the brushes contact) looked fine.

I can write a lot about alternators but it probably won't help you. 

What's humorous is that you said your 2004 Nissan has 51k miles and is only seldomly driven by an old lady.  The point is: the alternator has 51k miles worth of wear on it!  It's NOT slightly used.  It's time for the brushes to be worn out. 

Of course there are other problems that could have happened, but my point today is "normal" wear and 51k miles.  51k miles is not "low usage".

I'd really like to know what "failed across the board" means to you.  To me the term is usually related to money and probably gambling. 

If you're a DIYer, I encourage you to disassemble a dysfunctional alternator and see what's inside.  That's how I learned.  Only I have a source for alternator, starter and generator parts.  You might not know where to get brushes. 

-Engine overhaul mechanic and general automotive mechanic since 1972

It was jump started backwards and fried the rectifier bridge at some point. With the battery tender on it she may have been ignoring the warning light for awhile if she only did short trips. The tender would charge it up over night so the car might be good for 30 minutes the next day.

There could be many reasons why something fails too early,

Climate can play a part as moisture can build up in internal parts and cause corrosion,

Also if the vehicle is not being used on a regular basis small amounts of current will be being drawn from the battery all the time it stands there,

Alternators put out full power when the battery is quite flat but it is not good for them,

so a vehicle that hardly gets used will have this happen when you eventually do use it.

And also, in my job (Auto-Electrician) I believe that regulators on cars after 2000, became more like mini computers that the old style regs, There is nothing to stop manufactures making any computerised part fail, whenever they want to

We noticed that VW regs were failing at around the same time regardless of the mileage the vehicle had done

If the car was jumped stared the fusible link may have blown and the alternator can't charge the battery if the fusible link is blown. The fusible links are the larger fuses in the engine compartment fuse box. Some Nissan's have the fusible links on the positive battery cable clamp. 

Corrosion and age.  Or simply a bad product.

It’s 16 years old.  Age probably whacked the diodes or the internal regulator.