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these books, I also find no mention of it. But at seminars and lectures in differential geometry and mathematical physics, one very often hears the need for coordinate-free methods. It's a kind of article of faith which I think arose because of the extremely annoying "sea of indices" which confronted the reader of any book on differential geometry from the late 19th century up until the 1950s. Then some authors made an effort to get rid of the coordinate indices. I think it's the subscripts and superscripts that annoy people the most, not the coordinates per se.
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in my opinion, the reason for the absence of references for the "coordinate-free" topic is that this is part of differential geometry folk-lore. I've just looked through many of the 45 DG textbooks in my collection, and none of the authors seems to index this topic. Looking through the insides of
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Physicists still use coordinates all the time because they need to make practical computations. Pure mathematicians can mostly avoid coordinates by using the Koszul vector field calculus or fibre bundles.
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The one place where I see the issue in print is in
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