That was another one of those WTF moments, when he so often repeated this Sputnik moment that he would aspire Americans to celebrate. And he needs to remember that what happened back then with the former communist USSR and their victory in that race to space, yes, they won, but they also incurred so much debt at the time that it resulted in the inevitable collapse of the Soviet Union.
The wrongness, it burns. Like saying:
The British built the first operational aircraft carrier in 1918, but they also incurred so much debt at the time that it resulted in the inevitable collapse of the British Empire.
Sputnik 1, Earth's first artificial satellite, was a cheap and simple spacecraft designed to use the Soviet R-7 rocket already being developed as an ICBM to reach orbit before a competing US payload.
The R-7 rocket was not a trivial project. Still, we know that the Soviet Union developed about a dozen other ICBMs, plus many shorter range missiles and several dedicated space launchers, while maintaining enormous conventional military forces, so developing the R-7 was probably a fairly small slice of overall Soviet spending. The current versions of the rocket evolved from the R-7 ICBM still serve as a reliable and competitively priced launcher for Russian and commercial satellites.
What really caused the collapse of the Soviet Union? It was a polyglot multi-ethnic empire cobbled together by the conquest of what used to be other sovereign states during the 18th, 19th and 20th centuries. As such, it was always unstable, requiring the continual willingness to coerce the constituent republics whenever they wanted their sovereignty back. Ultimately, Russia made a wise decision to let the other republics depart in peace.
That aside, could Russia have survived as a single party state ruled by the communist party? Quite possibly, if the government had moved sooner and farther away from a command economy and more towards a market economy: as quickly, as say, China. The essential problem was that the Soviet command economy worked very badly and was falling further and further behind the west.
Not overspending on a grossly oversized military for the preceding four decades would have given them more time to make the transition. Not invading Afghanistan would also have helped.
After the part of the space race that the Soviets won they did have some expensive and ill conceived projects: the giant, underfunded and cancelled N-1 moon rocket and Buran/Energia. These were a great deal more ambitious than Sputnik and Vostok, and had they never been approved that might have allowed to Soviet Union's finances to hold out a bit longer. It's important to understand the scope of the problem. By 1985 the Soviet union had a hard currency shortfall of $20 billion a year..
At the official exchange rate, the Soviet Union is reported to have spent about $18 billion on Buran/Energia. That's a lot of money, but under those assumptions choosing not to build it would have postponed the collapse of the Soviet union by less than a year. The official exchange rate grossly overvalued the dollar value of the ruble, so the actual impact of making a better decision would probably have been a good deal less.
But that was decades later than the early part of the space race that the Soviets actually won.
So, to recap, according to Palin:
The former communist USSR had a victory in that race to space,
Big government command and control does everything worse than America's small businesses.
If the big government command and control USSR launched an artificial satellite before America's small businesses, it follows that they can only have done so by taking on inevitably suicidal levels of debt.
Ironically, Soviet aerospace was the part of the economy that functioned most like the West; the rockets were built by independent design bureaus that competed with each other for government business, much like western defense contractors. Many have successfully transitioned into joint stock companies.