"For Sarah Bakewell, the answer lies in the peculiar appeal, and the timeliness, of the philosophy that he espoused: existentialism. "
I have to disagree. Not that Existentialism is unimportant but I would assert that it was "something about France" in the sense that philosophers are highly respected and that they essentially are public figures akin to celebrities.
You could even say it shows "something about America" (or England) that other countries respecting philosophers as much as we respect reality TV show stars is considered strange and needing of explanation.
Actually, what Sartre called existentialism is the Asian way of life, at least in societies influenced by the original Hindu philosophy (Upanishads) and its Buddhist derivatives and finally the Gita.
I'm basing all of this on memory of something I've read long ago, but with enough google fu you might be able to find the references for these claims.
Some criticism: Sartre isn't as respected as the others in academic philosophy (in popular culture, it's another story). Heidegger was an unrepentant Nazi who turned on his Jewish mentor after Hitler came to power, and Schopenhauer was considered a hypocrite by Bertrand Russell.
Nietzsche is clean, and pretty much practiced what he preached.
What evidence is there that the Asian philosophies you mention were anything like Sartre's existentialism?
So, the seers of Upanishads and non-lamaist sects of Buddhism (Hinayana), being seekers after truth with emphasis on seeing things as they are have described most significant principles of philosophy of the mind and moral philosophy something about 6 century BC, if we consider philosophy of the Buddha as culmination of that epoch.
There is the famous two volume work by Radhakrishnan which gives obviously a Hindu centered overview of the philosophy of Indian civilization. All the fundamental ideas are there, including the notion that all phenomena are devoid of any absolute essence, are mere appearances to the senses of different aspects of what they call the whole - Brahman.
Some later Buddhist sects have rejected the notion of Brahman and maintain that the real nature of all phenomena is void.
To put it simple - Hindus got the fundamental principles right few thousand years before West.
The aspects of Eastern philosophy you mention seem more relevant to Western ideas like phenomenology, monism or pantheism. The identifying feature of existentialism, as distinct from phenomenology, is the focus on the inescapable responsibility that our individual freedom places on us to create our own meaning in life and accept our own ethical behaviour and value system. It is an individualistic and atheistic perspective. The phenomenological aspects are more psychological than the Eastern works you mention, which are more metaphysical.
You say "I am not sure that classification of philosophies and philosophers is of any use", but you raised the issue of the classification and its origin.
...I have nothing to apologize for.