I have always regarded as one of the fundamental shifts from classical physics to modern physics to the shift of emphasis from seeing a system described by forces, mass and velocity to seeing a system described by the conservation of momentum&energy (the concepts of mass&force receding into insignificance from the fundamental point of view), along with various requirements of symmetry/invariance.
It is a subtle shift, and it is easy to superimpose a modern view on the old-fashioned classical view.
However, if one does so, then we run the risk of not understanding what classical physical actually concerned itself with, nor how its perspective was internally consistent, but still obscured certain features that the perspective of modern physics handles better.
Take the case of the "principle of Galilean invariance":
I must confess that I haven't found this requirement stated in any pre-Einstein physicist.
Rather, to them F=ma was the fundamental law empirically verified to hold for material systems (systems that consisted of the same stuff over time).
Also, it was empirically verified that a system only could lose mass if some part left the system (mass conservation law, which I believe is the oldest conservation law)
From F=ma, it is fairly trivial to DEDUCE that there exists some sets of "equivalent observers", namely those that move with constant velocity to each other. For these groups, the force F acting on the object will be observed to be the same, since the accelerations are the same, due to the observed kinematic Galilean law that velocities are additive.
If we call one set of such observers the "true" observers, who deduce the ACTUAL forces on the object, then the other sets observe additional pseudo-forces, due to their acceleration relative to the set of "true" observers.
But at no point is this equivalent to state that classical physics REQUIRED the laws of mechanics to be Galilean invariant, they OBSERVED, or DEDUCED that they were. Nor, indeed did the idea of "absolute state of rest" lose its meaning, it was just that one couldn't deduce an absolute state of rest with the laws of mechanics!
This means that when the phenomena of electro-magnetic forces began to be studied, there was NO CONCEPTUAL CONFLICT with previous physics, rather what one discovered was that since the laws of electro-magnetism were not Galilean invariant, it followed that the state of absolute rest could in principle be deduced/observed by the study of electro-magnetic phenomena. Maxwell's laws were assumed to be valid for the absolute rest frame, and hence another observer's absolute velocity could in principle be deduced from HIS observed laws of electro-magnetism under the assumption that all velocities would, indeed, follow the Galilean empirical law of velocity addition.
This perspective is internally consistent, even though we now know that several of the assumptions are wrong.
However, as I hope I have shown, it wasn't because physicists previously were dumb that they didn't question their assumptions when the non-Galilean electromagnetic phenomena appeared; it was simply accommodated easily into their system of thought as some rather weird forces.
However, with the Michelson-Morley experiment came, one of the basic observational laws hitherto known failed, that of velocity addition, then something seriously wrong were understood to be the case.
However, Einstein's revolutionary thoughts, for example his fairly unique requirements of invariance, are by no means the only, or most obvious way to try and find resolutions to the problems at hand.
Others were tried out, most of them forgotten because at one point or another, they failed.
