Do research-only postdocs harm chances for tenure-track opportunities?

Suppose a person spends their PhD at an institute with a high ratio of PhD students to other students, resulting in a very low teaching load for PhD students, resulting in no or almost no teaching/tutoring. Suppose he/she spends years in postdocs, where each postdoc is funded by supervisor grants, and involves no teaching. She/he does great science but does not acquire other skills or experience.

Consider that tenure-track positions always require stellar experience in research, teaching and getting proposals funded — all of the above.

Is it harmful to an academic career if research is great, but all done leeching of other people's grants and with no teaching experience to speak of? If yes, how could one prevent or overcome this?