You can't just point to a list of inventions that came from research in space exploration - you also have to make a case that these inventions would have otherwise not come about, or at least would have come about much later.

I disagree. You would have to make the case that these inventions would come about without space travel - that there were alternative commercial interests and government incentives that would finance the necessary research.

But you're the one using these inventions as justification for future space exploration - shouldn't the burden of proof fall on the one making the claim?

Those inventions provide justification for space exploration - they establish this field as a tried-and-true way of generating very useful spinoff tech.

To counter this argument, the other side needs to justify that either those same inventions can be achieved in a different and more cost-effective way, or that we don't really need new tech (relative to its costs).