The definition of the profession though is honorable - there are certainly teachers that are not - and that do not go into the profession to make a difference in children's lives but just to draw a paycheck.
What does 'the definition of the profession' matter when it comes to evaluating individual teachers? Either you're a good teacher or not. All the definition does is allow a bad teacher to hide their bad and rely on the definition to make them look good, i.e. to lie/fake.
But my experience having two children in the private at one point and public currently school system is that the majority of teachers do really care and try to impact the students they teach.
Again, so what? If the majority of teachers were bad, that would not erase the possibility of teachers getting it right if they tried. Likewise, if the majority are good, that doesn't change anything about the bad ones who are bad, except maybe to hide them in the mix.
I think that teaching as a whole is an honorable profession both in the private schools (who actually get paid significantly less) and in the public schools.
Teaching is good from a certain POV, which is that it is good to transfer knowledge and skills so the general population grows more intelligent and responsible. From another POV, knowledge and skills should be restricted to experts so that they can make more money serving more people who are kept dependent on them.
In many ways, the second ethic is the only the current education serves. The paradigm of specialization and division of labor basically just promotes the idea that students should be prepared for specialized education after graduation. As a result, teachers are basically just helping to keep students ignorant with busy-work as they await expensive higher education.
Is that 'honorable?'