Are unimpressive grades in core courses an indication of not being competent for a career in the field?
You are overthinking it. First, if you complete your degree and get that first academic job, no one will care anymore what your grades were. 

Second, you seem to be implying that your learning ended in those subjects with the courses you took. I think that isn't very likely. 

In fact, for teaching purposes, it is sometimes valuable to have had the experience of struggling earlier, so that you have an appreciation for those students that don't find everything easy, or even trivial. You will probably be required to teach students who are not at all like those who go on to earn doctorates - i.e. not at all like you. Every student is different and the strugglers need to be taught along with everyone else. 

There are many reasons for mediocre grades, though yours are a bit better than that. I suspect that things may have been hectic at the time and there wasn't time enough to reflect and gain deeper insights. But that level of insight isn't forever closed off to you. 

As for the research, in many fields the research is so deep and narrow that it is almost necessary to give up a deep knowledge of other somewhat related, but not essential companion fields.