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Brace yourselves, this is going to be a wall of text, with a lot of supporting articles and videos.

The reason that they thought this was a good idea was because they had an ulterior motive of getting the Assault Weapons Ban (AWB) renewed in the US. Guns going to Mexico was merely the stalking horse used to achieve this. This was a coordinated, in depth, multipronged campaign aimed towards getting that one goal accomplished, with the US press more than willing to help them, both in building the case, and helping minimize the aftermath.

Let's go back in time to just after the 2008 Presidential Election - Clinton was named Secretary of State, and Holder was named Attorney General.

In February 2009, Holder stated in an interview that the President was seeking to have the AWB renewed.

In March, Clinton went to Mexico and made noises about how US weapons flowing into Mexico were fueling violence.

In April, President Obama and Mexican President Calderon appeared together and stated outright that the assault weapons ban should be renewed.

At the same time, the ATF released "trace data" showing that America was supplying the majority of weapons turned in for tracing to the ATF. The press made a big deal about this, though it later turned out that the numbers were fudged - even FactCheck got involved and there's several articles at the bottom of that showing how much this was in the news. Politifact also evaulated the claim and found it half true

The public wasn't having it. Polls showed that fewer Americans supported stricter gun control than ever Congress wasn't particularly interested in sticking their necks out either - they remembered what happened the last time the AWB was passed; the Republicans took control just 7 weeks later.

The NRA wasn't going to let this slide and made their displeasure known - every time the administration made a statement about guns, their predictions were proven right.

In May, Calderon appeared before a joint session of Congress urging a renewal of the AWB.

Things were going on behind the scenes in June:

In a statement Thursday, W. Larry Ford, ATF assistant director, said the bureau and Immigration and Customs Enforcement "are working to finalize a memorandum of understanding to maximize our joint effectiveness to combat violent crime along the Southwest border."

This went on throughout the summer and into fall withthe press providing articles trying to help the administration build the case. In October, the US announced that it would support a UN treaty to regulate the arms trade, a reversal of previous positions.

About this time, Operation Fast and Furious started. The objectives were clear - let as many guns into Mexico as possible and sit back and watch what happened. John Dodson later testified to Congress that for 10 months agents were told to stand down, and just watch straw purchasers buy firearms over and over again. He & his colleagues were directed by supervisors not to make arrests or interdict the weapons - even though they would show up at crime scenes days later. "Allowing loads of weapons that we knew to be destined for criminals [to walk] was the plan. This was the mandate." He testified that prior to this he had never heard of an operation that would "just let the guns walk" and when raising concerns he and his coworkers were told that they simply "did not understand the plan." Summing up, he stated "I can not begin to think of how the risk of letting guns fall into the hands of known criminals could possibly advance any legitimate law enforcement interest." The result was lots and lots of bodies, hundreds in Mexico alone.

In hindsight, and viewing the greater picture starting from just after the election, the macabre goals become clear: The more deaths, the more ammunition the administration had to push for additional gun control.

Thousands of guns were lost, and nobody gave a shit until Border Patrol Agent Brian Terry got killed in December 2010. Shortly after his shooting, the ATF traced the gun; it turned out that it was from the F&F operation. This sent shockwaves up the chain of command. The administration shit themselves and tried everything possible to mislead, misdirect, and cover up what had happened, and (at the time) largely got away with it. Concerned about how it was handled, a number of ATF agents went on CleanupATF.org and posted about F&F anonymously. This started the ball rolling; as word spread people wanted answers for how he was killed and where the gun came from.

Issa started an investigation. Surprisingly, one of the ATF Agents involved in the Fast and Furious was also WTF about the directives he was given and wound up getting crucified by the agency when he blew the whistle to Congress. Here's part I of the F&F report, and it goes into detail about how it was a complete clusterfuck. The administration also stonewalled, slow rolled and delayed documents from the investigation to the point where Holder was held in contempt of congress for failing to turn over subpeonad materials.

Internal ATF emails showed agents shopping for anecdotal stories to help build a case for "Demand Letter 3" - multiple long gun reporting, but this as sole justification for the operation falls flat.

Sen. Charles Grassley, R-Iowa, is investigating Fast and Furious, as well as the alleged use of the case to advance gun regulations. "There's plenty of evidence showing that this administration planned to use the tragedies of Fast and Furious as rationale to further their goals of a long gun reporting requirement. But, we've learned from our investigation that reporting multiple long gun sales would do nothing to stop the flow of firearms to known straw purchasers because many Federal Firearms Dealers are already voluntarily reporting suspicious transactions. It's pretty clear that the problem isn't lack of burdensome reporting requirements."

The Terry killing completely sidelined gun control efforts from that point on; the Administration was in CYA mode and hid what had happened so successfully that when Gabby Giffords was shot at a rally the following month (January 2011), gun control groups and the press both wondered why the President wasn't taking steps about gun control. We now know that they were trying to manage the F&F fallout.

Everything possible was done to shield the administration from blame and criticism. Holder blamed Bush, trying to link the Wide Receiver Operation (that ended 18 months prior to F&F) to his failed operation. President Obama stated that F&F was a continuation of Wide Receiver during a Univision interview - Politifact rated this as false, but this lie continued to be repeated. In one of the most disgusting excuses for journalism I've seen, Katherine Eban wrote up a hit piece in Fortune that completely absolved the ATF & administration of any wrongdoing, pretending the gun walking wasn't intentional, casting Issa as a villian for investigating, and blatantly attacking the character of the whistleblower. This article was heralded far and wide as proof that nothing untoward had happened - this was debunked in detail but the damage had been done and little attention was paid in the run up to the election.

Edit: In a last ditch effort, Obama cited executive privilege and hid 15,000 documents from Issa's investigation. Anyone raising a stink about this was ignored.

Once the President was reelected, it no longer mattered, and slid completely off the radar.

I don't remember exactly, but IIRC nobody involved lost their job over this. Certainly nobody responsible went to jail. Many of those involved were transferred and then promoted, but ultimately nothing was done.