Newtonian physics is a special case of relativity when you keep relative speeds slow.
We don't know how quantum mechanics fits in. There is a known hole in current physics. However the size of the hole is small.
My point with Newtonian physics is that at the time, it was The Physics. According to observation capable at the time, it seemed to make sense, with a few little oddities.
To them, they thought they had a small hole in current physics. Does that sound like your "small hole" in current physics today?
You believe we are knowledgeable, for we posses knowledge our ancestors do not. You believe we are right, for our observations tell us so. Yet how will we observe the universe 1000 years from now? 100? What tools will we have?
Place yourself 1000 years hence. Our ancestors will look back, and see our current physics as we view Romans babbling on about The Elements and aethers. Yes, to our descendants, our science is absurdly simple, filled with holes, broken, the list goes on.
We are not even remotely sure if space/time is consistent outside of our local region of space!
Back to your Newtonian statement, what if all of our current physics was thus? A viewpoint which only worked under certain circumstances. From where I sit, this is the reality.
We work with what we have, but must keep the knowledge of our massive ignorance in play too!
I come back to this: even if we are wrong, whatever is correct still needs to fit all the observations that we have done to date. I don't know what we will know in 1000 years. Euclidean geometry is over 2000 years old and still considered correct (though we have expanded on it a lot - see in particular non-euclidean geometry)
As bluGill says, according to observation right now it _still_ makes sense. With a few little oddities.
Your view appears to be one of relativity refuting, rather than further confirming (as far as the scales studied at that time) classical physics.
You cannot compare it to something allowing FTL. There's a major difference between "we don't know everything" and "everything we know is wrong".