In that conversation, nobody flat out says "the science is settled" because it is easy to explain the science behind the earth being round. No, "the science is settled" is only ever used as a way to end a discussion without explaining the details of one's position. Not that they're necessarily wrong, but just that they don't know the answers. It is always an attempt to end a conversation on an authoritative basis.

If you get drawn into a debate about whether the earth is flat, you're the sucker and are being made fun of by the person "debating" you. And retorting that "the science is settled" would be a ridiculous response.

In serious debate, "the science is settled" is the intellectual equivalent of "because I said so". But I don't believe it comes up often in serious debate, it's more likely to happen when two parties that don't really understand science are having an ideological debate that they pretend had to do with science as a rhetorical device to gain legitimacy.

It's better to use different language, like saying how we're exceedingly confident that the earth is round, rather than it being an absolute. Sort of like our confidence asymptotically approaches absolute but never actually reaches it.

This is similar to Hume's argument for our belief that the sun will rise tomorrow. Obviously, we all believe (even flat-earthers) that the sun will rise tomorrow and we have much science to indicate this will continue to be the case for a long time. Still, that doesn't mean we can be 100% confident this is the case. It's frankly just exceedingly difficult to be 100% absolutely sure about anything, unless it's first order mathematics.

The Earth might not be flat, but it isn't round either. That one is a lot more wrong than the other matters little if you're trying to land a spaceship.

No it’s never context dependant. In regard to flat earth, the scientific response is simply that the evidence seems to contradict that theory. If some people think it’s still a valid line of inquiry, then they’re perfectly entitled to pursue it. Science doesn’t dictate what is and is not a waste of time. If some people wish to assert that flat earth theories are definitely true, then that’s fine too. It’s not scientific, because science doesn’t have any mechanism for determining something to be definitely true. If some people want to claim that flat earth is likely true, that’s fine too, but it’s also not a very defensible scientific conclusion either, because the scientific evidence weighs heavily against that conclusion. But saying that something is definitely not true is equally anti-science, because science doesn’t have a mechanism for determining something to be definitely not true either.

The problem that people have with this tends to be that when they notice somebody being “wrong” about something, they want to be able to conclusively prove that the other person is wrong, and that they are right. Especially if it’s in regards to an issue they consider important. If you think that the scientific evidence supports your position in a situation like this, the temptation is to claim that science unequivocally proves you are correct. It is however, an outright abuse of the scientific method.

This sort of situation has become very common place. When science communicators are faced with anti-scientific assertions of absolute truth, they’re not inclined to acknowledge scientific uncertainty, and the limits of their own understanding. Sadly this has created a culture of anti-scientific science worship.