Paul: Almost entirely agree. Where we may differ is in the level of understanding required. One needs to know what those words mean but one does not need to be able to conduct original research in those fields. What is required lies somewhere in between. In my opinion, anyway.
That’s a major reason why I recommend MTW. It not only defines those words, it gives a relatively gentle introduction to what they mean and how to use the concepts in practice. Though, to be fair, Lagrangian dynamics is (IIRC) treated in the advanced track and can be skipped on initial study. An STEM undergraduate level of group theory is undoubtedly very useful but may not be strictly necessary. Again, IMO.
It is not unusual for pedagogues to disagree on details. What is unusual is for them to agree on all the details.