[chombier]

Of course you are correct, even though one could argue the linearity of differentiation is a property you can obtain by differentiating in coordinate charts, where the reasoning is still valid.

In any case, it is probably a good thing to get a good intuition of what differentiation and derivatives are in the vector space setting before digging into differential geometry.

[4ad]

Not all vector spaces are manifolds (in fact most aren't), and you don't need charts to define differential operators (just see functional analysis).

[chombier]

I most definitely agree with you, however this brings us even further away from the author's setting.