Everything up to here is fine. But this one (and consequently the next), why do you think that?
Why would corporations finance interstellar travel anyway? There's no profit in that.
(Personally, I still think the most palatable way to travel in that environment is to make your ship so damn good that you don't even care if it's getting anywhere.)
Shipping.
That's exactly what the Nostromo was: a shipping container. They were hauling goods from one planet to another for some (profitable) reason.
What we know now, however, is that a great deal of that could likely be automated / not require a crew. But even then: the Nostromo was massive with what could amount to a skeleton crew (half a dozen?). Container ships even today have more than that on them.
If they won't, then nobody will. The most you'll get is a few state-prestige projects which fly around a tiny handful of people at most before they get bored and move on. This is probably the most realistic outcome but it doesn't make for much of a sci-fi story so it's better to take some liberties and assume that commercial exploitation of space flight might somehow be possible. Once you've accepted the possibility of space commerce for the sake of the story, you can pull publicly funded space travel into your story as well if you wish.
You could also use war as a justification, but the viability of the sort of interstellar war that involves spaceships of people flying around (rather than the use of long-range weapons of mass extermination) likely suggests the viability of interstellar trade or at least resource exploitation.
Churches? To spread the good news to alien worlds?
Resources? Should we find planets with lots of them but a surrounding environment that is hostile to human life, there would be enough demand for mining and automated manufacturing with cargo ships going back and forth. The industry would likely love being permitted to pollute those planets at will since there would be no biosphere that could be harmed.