Historically, why did the US station their nuclear weapons in Germany for delivery through German pilots?

This is intended as a historical complement to the technical answers.
I'm aware of the treatment it is liable to receive.
I still believe it is worth posting.
Hopefully people will take it as a useful complement to the more pragmatic detailed material in other answers.

I write this in 2020.
75 years ago the US was just concluding part 2 of the-war-to-end-all-wars.
This had been intended to be a 1 part war but "things went wrong".
The thought of part 3 occurring at some future date was not to be countenanced.

There were strong opinions expressed in the US hierarchy that Germany should be not just demilitarised but deindustrialised - the Ruhr stripped of most of its factories and the nation rendered essentially pastoral. While this was neither the major or ultimately prevailing decision, it made sense enough at the time. For those doubrful of these ideas or not familiar with "The Morgenthau Plan" a skim through this material should be instructive). (Morgenthau was Roosevelt's Secretary of the Treasury.)

Nobody, and that may well literally be nobody, would have imagined that well within 100 years the land of Naziism, the Third Reich, Lebensraum, concentration camps, ethic cleansing before it was a thing and so much more would ever take it's place as a willing and capable democratic partner in Western Europe. The idea of the US providing Germany with nuclear weapons for (potential) delivery by German pilots would have made 'Dr Strangelove' seem mild.

And yet, here we are.

In part 1 of the war-to-end-all-wars (to be continued) Canadian physician Lieutenant-Colonel John McCrae wrote a poem. We know it as "In Flanders Fields" It contains a passage intended as a challenge that instead became of such immense prophetic import that we read it and miss both what he was asking and what has been achieved.

Take up our quarrel with the foe:
To you from failing hands we throw
The torch; be yours to hold it high.
If ye break faith with us who die
We shall not sleep, though poppies grow
In Flanders fields.

Somewhat to my surprise, I often cry when I read this poem.

When McCRae wrote those words his idea of "take up our quarrel with the foe" meant just that. He would very much have agreed with the thoughts expressed in The MLF lullaby by the great Tom Lehrer - as noted and link supplied by Moshe Kohan in a comment.

And yet. Here we are.
After WW2 the threat posed by the Soviet Union, and the need to aid Europe lead, in an inseparable complex mix to "treating the vanquished well by any past standards, the Marshall plan and the cold war.

McCrae's challenge needed addressing.
The idea of round 3 of the war-to-end-all-wars popping up again as soon as the German's recovered their strength was beyond unacceptable. The Soviets provided the spur, if any was needed. Japan got massive US industrial advantage in the form of the one man army W.Edwards Deming. Germany got among other things, the Berlin Airlift (June 1948 May 1949) and support as a nation against the perils of the day. And it went from there.

McCrae's challenge was answered in a way which would have been beyond belief in 1945, and hardly more believable 3 years on as the US rallied to the support of their ally. (!).

What we see now is the logical conclusion of the path chosen then. Certainly, political relationships between German and the rest of Europe may often be "a little frayed". Even US - German relationships are not always as amicable as some would hope. But US nuclear weapons in German aircraft with German pilots (no matter how the actual lines of control may actually work) follow a progression expressed as a hope by a WW1 surgeon, fostered by Stalin as he sought to divide and conquer immediately post WW2, and ongoingly encouraged by the ongoing enmity between "The West" and The Soviet Union.

So, Why?
Arguably, no greater trust could be expressed in ones allies than to arm them with your own nuclear weapons, able to be delivered by their own pilots. It's logical. :-)

Someone better versed in the relevant history and politics might give you a more precise answer, but being able to read German sources is at least somewhat helpful. In short, there was also a lot of power politics going on. See for example this article (in German).

Essentially, in the 50s and 60s, everyone wanted their own nukes. That included Germany's chancellor Adenauer and in particular his minister of defense Franz Josef Strauss (probably the quintessential example of a right wing hardliner in past WWII German politics even to this day). What the US feared was that Germany would develop nukes on their own or in conjunction with France and that this would then allow them to declare themselves neutral or even strike a deal with the Sowjets, effectively ending the US presence in Europe. In theory there were treaties against this, but then again the same was true regarding remilitarisation after the first world war.

The solution to this was nuclear sharing. Germany gets access to nuclear weapons, but only as long as they stay in the NATO and allow the accompanying US troop presence. There is also the neat side-effect that they have to buy US equipment to use those weapons, which is a part of the current controversies. In general, the topic of US nukes comes up in German politics again and again every few years, mostly as a proxy for US bases in general, as no one seems to really be satisfied with the status quo. But there seems to be no solution which won't either offend the US or a considerable amount of German voters, so the usual result is inaction.

False sense of security

Mainstream wisdom during Cold War was that if ever that war becomes hot, Soviets (and Warsaw Pact in general) are bound to invade Western Europe, before substantial reinforcements could come from CONUS . Soviet military did have various contingency plans, some were almost purely offensive ones like 7 days to river Rhine, some were more defensive. In reality, Soviet political leadership (with advice from allied countries) would have to make decision, and they were more defensively oriented then previously thought, fearing that Americans will strike first .

Anyway, main problem in strategic thinking for NATO side was first use policy for nuclear weapons. If the Soviets do attack and invade Western Europe, but initially refrain from using nuclear weapons, should NATO use them first, and what would be the threshold. Question is not easy, because if NATO starts dropping nukes (even only tactical), Warsaw pact would likely retaliate in kind, leading to global catastrophe. On the other hand, if NATO does not use them, what is exact purpose of NATO ? French already had their own nuclear doctrine, suspecting that US (or UK) would not unconditionally defend Europe.

To resolve this crisis, but at the same time to dissuade NATO countries and other allies from pursuing their own nuclear programs, US decided to "loan" to some of them surplus nuclear weapons. This goes especially for countries like Germany, Italy and Turkey, which with their sizable military budgets could afford to develop their own nuclear missiles, thus increasing proliferation but also weakening position of US as superpower defending Western world and member of exclusive nuclear club. In practice, all this nuclear sharing amounted to delivery of B61 free-fall nuclear bombs.

Why is this false sense of security ? As mentioned before, nuclear bombs in question are free-fall bombs. That means that their range is limited by range and survivability of an aircraft carrying them. It is highly unlikely such weapons could be delivered to Soviet strategic targets trough dense air defense and enormous range using conventional fighter-bombers available to NATO allies (Tornado, F-16, F/A-18 etc ...) . Even as tactical weapons, dropped near the front, their usefulness would be limited - they may kill some Soviet troops but also devastate local population in presumably NATO countries being overrun by those troops. And of course, there is a question of retaliation.

So what was the point ? We trust you enough to give you the bombs, but we know you would not use them. Politically, it was expedient for politicians in Germany, Italy or Turkey to emphasize that their country now has nuclear weapons (well, sort of ) therefore increasing prestige of the country and their own popularity among population. Indeed, Americans trust us so much that they gave us their nuclear weapons. But from strategic and military POV, US knew that chances of those weapons being used is almost null, even in shooting war. What mattered to US the most is that with existence of nuclear sharing program, Germany and other countries would not embark on their own independent military nuclear development, therefore staying firmly in NATO camp and depending on US.

As your article itself states:

On the record, the Germany government only admits to being part of what is officially termed a "nuclear sharing agreement."

In essence, the nuclear sharing agreement provides for member states of the military alliance without nuclear weapons to partake in planning and training for the use of nuclear weapons by NATO. Additionally, officials argue, it allows for their views to be taken into account by nuclear-capable countries, including the US.

Basically, the cost paid by non-nuclear members of NATO for having a say in making and implementing policy and decisions in regards use of these weapons is the commitment to deliver them when called upon.

Any member state wishing to sit at the big boys table has to make an adult commitment to the alliance.

Full stop.

During the Cold War, there were always doubts if extended deterrence would hold. Would the US put Washington at risk to defend or avenge Bonn or Frankfurt?

So NATO needed powerful signals that they would stand together, even at the expense of operational efficiency. Compare the multinational battalions in the Baltics today -- those cannot stop a Russian invasion, but they make sure that everybody gets involved.

• It means that any attack on German tactical air power is at the same time an attack on the delivery system for US nuclear weapons. Theory and practice of nuclear deterrence is both controversial and complicated, but there is widespread agreement that an attack on nuclear command-and-control infrastructure and delivery systems is a highly escalatory move. A hostile (nuclear or non-nuclear) power will not take this escalation lightly.
• It means that in the event of a full-scale nuclear strike in Europe, the German government cannot equivocate and blame others. They (the Chancellor who will be commander in chief at this time) must get off the pot and join -- or not.