They're incredibly important, but they shouldn't be touted as facts until there's some consensus that they are facts
We use scientific models because they are useful not because they are fact. Even if our model does not always match our observations it can still be a useful model in some circumstances. It's very important when talking about science that we understand that it isn't fact. It is a model. It might be a good model. It might be a bad model. It might be a useful model for our application. It might be a poor model for our application. That's it.
When you see a headline that says "X is Y" and it's about a scientific model, what that means is "If our model is consistent with the real world then the model predicts that we should find that X appears to be Y in the real world". But that's a bit too long to put in a headline and you would have to do it for every scientific conversation.
It is extremely unfortunate that on the first day of junior high school when you start doing "proper" science classes that they don't sit people down and explain what a scientific model is. Mostly I think it's because the teachers don't know, because their teachers never taught them.
This is a sincere question, and I care to hear your opinion: do you believe that a reader of phys.org would need more than a glance at the article to know this is a discovery about the mathematics of general relativity, not an engineering breakthrough in the creation of a scifi-like hyperdrive?
That's an overstatement. Measurements are more important than theory. We cannot form theories without measurements, and yes theories inform our measurements, but science always begins and moves forward with measurements.
Edit: Unrelated, but I see you’re a fellow Greg Egan fan, good to meet you! Planck Dive has to be one of my all time favorites.
Same with GR: it was a purely mathematical construct for a while until we could test it, but there were too many theory clues that it must be right.
And this is not selection/survival bias: it is extremely rare for humanity to find a robust mathematical construct for which we can have a high degree of certainty that it describes the universe well. In the few cases where we have had that certainty, due to mathematical proofs in seemingly disjoint fields, we ended up being right.
To push it to stuff like super strings and quantum gravity: few respected scientists would claim that their pet mathematical construct is correct, but many of them will say "the vague commonalities between all these diverse and seemingly unrelated mathematical constructs definitely point to an underlying fundamental construct".
Double-plus goodthink, comrade!