The scientific method is based on a logical fallacy

But tossing out gravity is also being proposed just as you seem to think should be done. There are some suggestions for a replacement.They are exactly following the evidence but haven't figured out where it is leading them yet.Keeping General Relativity and it's newtonian subset seems like a reasonable thing to attempt to do since it is so well supported in other ways. But even though it is very, very reasonable other things are being considered.

No, how foolish would that be! We don't overturn a well supported theory based on a mystery. We have to dig into the mystery first.There are multiple competing explanations for the anomaly. If some turn out to be right gravity will be falsified. It will almost certainly be falsified in the same why the general relativity falsified Newtonian mechanics though. Even falsified as shadings. What world do you live in? We have not, by any means, falsified anything yet. We have some interesting observations that, if correct, will probably lead to some interesting science. It is waaaay premature to leap beyond the "mmmm that's interesting" (from Asimov) step yet.At the edge of the science at any given time there are lots of "interesting" things. Some have more potential to shake things up but all are what is fun about science. However not all will survive.Stay tuned to this channel.

At what level of education? Newtonian physics can be taught as "true" to one set of students. Then they get the WOW fun when they find out about relativity. They have moved the introduction of relativity down about 4 years in the educational system here since I was in high school (yes it had been discovered then ). But you can't teach it all at once.I am at a loss as to how you could operate in the real world thinking as you do. You take small steps to understanding. Is this like Jeopardy? You ask the question to the answer?

Nope not at all. Read more s l o w l y. A difference being that they don't have any existing evidence based theory to work on. Instead of having to deal with a mystery they have to ignore a HUGE amount of very non-mysterious evidence.Physics 101 is not interesting here. Don't pretend.