The Rust Belt has famously been in decline for years. Why did the government let outsourcing happen instead of trying to keep those jobs in the Untied States?

One thing that's fairly important to keep in mind is that the Rust Belt was first threatened not by American-owned business outsourcing labor, but by American products not being competitive with foreign corporations.

American manufacturing companies - most notably, automobile makers, enjoyed a good ~15 years of prosperity and limited competition as Europe & Japan rebuilt post-WW2. By the late 60's and early 70's, European car manufacturers had caught up while Americans had grown complacent and their cars developed a reputation for being less reliable. The oil crisis of the 70's really exacerbated the problems - Americans wanted the cheap and fuel efficient German/Japanese cars that the big 3 were not producing. That perception and demand change was fairly catastrophic for the big 3. (see here ).

It is also very worth stating that the 70's also marked a pretty big turning point in environmental policy within the United States, with legislation like the clean air act and efforts to address pollution by manufacturing. Cleveland's river fire was pretty infamous. That represents another economic and legal pressure that US-based companies face that developing nations tend not to, which would become huge in the 80's - 00's with China's rise.

By the 80s, Reagan did start a trade war with Japan to protect the American companies, pressuring them to voluntarily limit auto exports, and then later issuing tariffs aimed at the emerging Japanese electronics (Sony, etc).

That trade war with Japan was successful - it curbed Japanese dominance somewhat and was a factor that led to their 'lost decade' in the 90s.

But outsourcing by American companies - largely to Mexico - really picked up in the 80s. The American brands couldn't consistently beat the Japanese & German brands on quality or reliability, so they tried on cost - and labor was the biggest lever to do that. From a policy perspective, what was the US to do? They could be more protectionist within the US, but that doesn't help all the other markets they were trying to reach.

NAFTA was signed in the 90's, making it easier to outsource to Mexico. The perception was is that any loss of outsourced jobs would be offset by cheaper goods and increased productivity elsewhere. This turned out to be (very) true - but what policy makers failed to really account for was the (very) disproportionate economic impact by region.

The Gingrich-lead Republicans were philosophically opposed to the types of large-scale safety nets, retraining, and subsidization that would have helped the region - but there was no political consensus to do anything either proactively or reactively.

Tl;DR: Ultimately, the economic forces (resource extraction / environmental, labor costs, and automation) have long been stacked up pretty hard against US-based manufacturing. Most policy makers have implicitly or explicitly recognized this, but calls diversify the economy of those regions (and to make a national-level investment) have not been politically viable.