Let's look at the closest thing we have to space exploration, the era of colonization. It coincides with the rise of capitalism and not coincidentally. The East India Tea corporation which basically ruled India was a state run corporation. The State was in charge of it, but it paid out profit to private investors (in addition to putting money in the state treasury) -it was the definition of crony capitalism, because it was ensured a monopoly and run by the government itself. One could very easily imagine a corporation such as the East India Company subsidized by the government (or even un-subsidized) getting into space exploration for the mineral rights alone.
Space exploration may be the magic bullet the Karl Marx didn't anticipate. Marx identified two major problems which made capitalism doomed in the long term. 1st that it's an inherently antagonistic system between the workers and those that control capital. Now all class systems are inherently antagonistic so I'm not sure why Marx believes this would doom capitalism. Well, to be fair all those other class systems did vanish around the world --to be replaced by capitalism. The second is more worrisome. A capitalist economy must grow faster than the population, it must expand, it must open up new markets or it will die slowly. It will return to that adversarial relationship and those with capital will use capital to create more capital by extracting wealth from those who have less. This happens when economic growth is slow, because capital, like power, naturally consolidates. When capital is growing the consolidation is offset by the growth. Picketty's great book elaborates, Capital in 21th Century. However, if space travel (and thus commerce) becomes a reality than economic growth is limitless as space itself. In your face Karl Marx.
Now, you suggested scenario of global debt would never happen. What would happen would be closer to the 2008 collapse, mass bankrupcies, the debt would be worthless and the capital would simply disappear.
This would result in high poverty rates, possibly the collapse of the banking systems, maybe fall of governments themselves --not a space race.
I think the East India Tea Company scenario is more plausible. A government (Like the US) is debt riddled, but not yet bankrupt or collapsed just spending all their government budget on financial services to the point that in a few years they won't have enough to pay the interest rates and they know it, grants a private company exclusive rights to space, passing off the all the risk of capital to that company but in return enforcing a monopoly. That would be an example of a government leveraging it's power into capital (crony capitalism) which is as old time and much more believable. This way the the company makes money, and the government basically extorts them and their competition, basically the whole space sector, to pay off its debt. This is if you want the financial crisis leads us into space version. I think private companies taking the initiative to go into to space and competing with each other is more likely.