@layman,
1. I don't think that many modern physicists are openly challenging and rejecting SR. You can find them on the internet... but then again you can find people openly challenging the moon landing on the internet, and people openly saying there are ruins of ancient cities on Mars. You are Googling for what you want to find and finding it.
2. Of course, if new experiments show that SR is not applicable, or if new mathematical models are shown to be better, then the scientific establishment will embrace new theories. This has happened before (i.e. with Einstein and Bohr).
Of course it is the scientific establishment that is in the best position to judge when this has happened. The people who have the most expertise on Physics are the people who best understand. This judgement will be made by experts, not by random people on the internet.
3. The scientific establishment has been very successful at rejecting theories that don't pass the grade, and embracing new theories (such as Relativity and Quantum Mechanics) that are better than existing models.
4. When there is a dramatic new change to the understanding in Physics (as embraced by the scientific establishment), it generally advances... rather than overturns... existing theories. Einstein showed that Special Relativity was an expansion of Newton's laws. This isn't always the case, but generally it is.
Physics students still learn about Carnot and Newton because Carnot and Newton are still valid at the limits where we live. Part of what Physics students learn to do is to show at what limits modern physics is equivalent to classic.
Classical physics was embraced because it did a very good job explaining measurements that were possible in previous centuries.
I expect if, and when, there is a new theory that replaces Relativity, it will likewise be an extension to Relativity rather than a completely new direction. There is a reason that Relativity does such a good job at explaining our current observations.
5. If Science advances, I will be happy to advance right along with it. But science has never advanced at the fringe.
Scientific advances, even dramatic ones, have always been done by the scientific establishment. There is a reason for this.