To butt in, I doubt your interlocutors were attempting to argue this point; they seem like they were having more fundamental issues. But your original argument does seem to be a bit confused.
Induction fails here because the inductive step fails at n=2. The inductive step happens to be true for n>2, but it is not true in general, hence the induction is invalid. The point is, rather than “you have to check n=2” or something similar, all that’s going on here is that you have to check that your inductive step is actually valid. Which here means checking that you didn’t sneak in any assumptions about n being sufficiently large. What’s missing is not additional parts to the induction beyond base case and inductive step, what’s missing is part of the proof of the inductive step.
To butt in, I doubt your interlocutors were attempting to argue this point; they seem like they were having more fundamental issues. But your original argument does seem to be a bit confused.
Induction fails here because the inductive step fails at n=2. The inductive step happens to be true for n>2, but it is not true in general, hence the induction is invalid. The point is, rather than “you have to check n=2” or something similar, all that’s going on here is that you have to check that your inductive step is actually valid. Which here means checking that you didn’t sneak in any assumptions about n being sufficiently large. What’s missing is not additional parts to the induction beyond base case and inductive step, what’s missing is part of the proof of the inductive step.