Differential geometry textbooks that define a smooth structure on $M$ as the set of all smooth real valued functions on open subsets of $M$

Question

I am looking for good textbooks on differential geometry that would consider the set of all differentiable/smooth functions $U\to\mathbf{R}$ for all open $U\subset M$ to be the main description of the differentiable/smooth structure on $M$ and would use it systematically. Any suggestions?

So far I've only found this one: Shlomo Sternberg, Lectures on differential geometry (1964).

The other textbooks I've seen so far define differentiable/smooth structures on $M$ using:

• coordinate systems or atlases on $M$ (Marcel Berger, Dubrovin -- Novikov -- Fomenko, Serge Lang, Michael Spivak, Frank Warner),
• local parametrizations of $M$ -- "inverted" coordinate systems (do Carmo -- Flaherty),
• embeddings or local embeddings of $M$ into $\mathbf{R}^n$ (John Milnor, James Munkres, Michael Spivak),
• pseudogroups of transformation on $M$ (Kobayashi -- Nomizu).