Change is hard. When computers were first introduced to schools there were similar arguments and complaints. When calculators were brought in instead of the slide rule we heard it would be the end of actually "knowing" math. Teachers who are great find ways to use technology and tools available to inspire their students and create engaging lessons. This teacher just sounds resistant to change.
2. Calculators do make people worse at math. I saw this first-hand with my son. As he went through school, they used calculators for everything. Students began to rely on the calculators for simple arithmetic and took the answers provided as gospel, not realizing that garbage in = garbage out. He, along with many of his peers, never really developed the fluency in math to say to themselves "hmm, that answer looks suspect. Let me try again."