Yes, you've got the core of the scientific method right in hand. You use math to formalize some model that explains both old and new observations, then you extrapolate using the math to make predictions, then you weed out all the failures with the next set of data. Everyone in physics does understand this, except maybe the string theorists, who went off the rails some time ago and IMO are an embarrassment to the field now.
But the math is the tool that lets you make that prediction, and it's not enough for the current model to be "wrong", you have to have an alternative that successfully predicts new data. That what all the internet cranks and crackpots seem to miss: yes, fine, you can contrive limitless alternative explanations for the existing data. So what? It's all about the predictions, and those are usually meaningless without the math to back them up, to set bounds and make them falsifiable.
The inability of GR to give a good model at universe-scale is hardly new - Einstein himself identified the need for a cosmological constant quite early on. The mystery has only deepened since - but that doesn't necessarily mean anything's wrong with GR, only that there's more to learn (dark energy is easy to describe in GR terms, but GR gives no explanation of why it's there in the first place).