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Yes, but first things first. When the audience doesn't have a
firm grasp of what a vector is, or how a vector field differs
from a vector space, you can't start with tangent bundles.
In fact a tangent bundle is defined in terms of tangent spaces,
so it makes sense to do tangent spaces before tangent bundles.
There is a rather general pedagogical principle that says "concept
before name". You can teach people pretty much everything they
need to know about tangent bundles without ever using the word.
One is free to dismiss it if you like, but when you are doing things like GR it really is a useful idea.
That's ambiguous, due to a dangling antecedent. Does "it” refer to rooted vectors, or to fiber bundles?