Does it (or something like it) exist, as a value or as a goal or whatever it is, in other schools of Buddhism (e.g. Theravada, or ...)?
Yes, it exists in Mahayana in general. From a Mahayana viewpoint, the mind of a buddha is free from conceptualization, and is effortless and free from intentions. (at least intentions that are conceptual consciousnesses, if not the omnipresent mental factor of intentions).
This effortlessness is often translated as naturalness, spontaneity, etc. In Chinese, the term is Tzu-jan (or ziran).
But while it is relevant on a soteriological path in Zen, it is not relevant in this specific way in other traditions or practices, to the exception of Mahamudra, Dzogchen and Tantra in general.
In China, it was a central concept to Confucianism and Daoism before Buddhism was introduced. It then became significant in Buddhism as well. The three teachings (Confucianism, Daoism and Buddhism) all relate to this effortlessness / naturalness but they vary greatly in terms of soteriological paths.
Whether we speak of Confucianism, Daoism or Zen, the idea is to act in accordance with circumstances, as opposed to acting in accordance with abstract principles that are general and not fitting particular circumstances. The following koan conveys this idea:
A monk asked Seigen, “What is the essence of Buddhism?” Seigen said,
“What is the price of rice in Roryo?”
According to commentaries, the meaning is that when rice is scarce, the price is high. Similarly, the essence of Buddhism depends on contexts. A Dharma practitioner must be adaptable and spontaneous/natural.
In the Chuangzu (a Daoist text), it says:
When a man shoots an arrow to win a tile he is skillful. But if he is
trying to win a silver buckle he starts getting nervous. And if he’s
competing for gold he almost loses his mind. His native skill is the
same in each case, but because he has something to lose, he overvalues
the external. Whenever the external is prized, the internal gets
To simplify and resorting to an analogy: From a Confucian perspective, in order to play piano "freestyle", playfully and effortlessly, one must have trained before, abiding by the rules, rehearsing classics, etc. On the other hand, the Daoist approach is more like this: From the start, one has to drop off one's intentions and opinions and not try to control ways and outcomes for, by doing so, he is getting in his own way. One already possesses "the native stuff", he just has to drop off whatever is superfluous. By dropping off, nature reveals itself. (that might echo Dogen's way of characterizing his satori experience as a "dropping off of body and mind")
Zen is very complex and multi-faceted. The so-called sudden teachings are closer to Daoism while the so-called gradual teachings are closer to Confucianism in its approach of rules, classics and traditions.
In terms of soteriological path, what is unique to Zen is that Zazen is to be taken in relation to the whole of Buddhist teachings, especially Mahayana. Some teachers are more or less study-oriented (like Zongmi, Yanshou, Chinul, and even Dogen, etc) and others are not.
I would advise the following books:
- Effortless Action: Wu-wei As Conceptual Metaphor. By Slingerland.
- Yongming Yanshou's Conception of Chan, by Albert Welter.
- Collected Works of Chinul, Robert E. Buswell Jr.
And a few valuable Daoist books that are more accessible than the Tao Te Ching. For instance, the Chuangzu and the Liezhi.
Why spontaneity is not relevant on a soteriological path in other traditions or practices?
From the 8th century (Kamalashila), Tibet sought to go back to the source (Indian Buddhism, not Chinese or Korean) but it took them centuries. In the 10th century, Atisa was summoned from India to Tibet to help revive Buddhism after it has been persecuted. Before even reaching Tibet, he sent the Tibetans the very first Lam rim. Lam rim is a textual genre that means: gradual path. The cultivation became a matter of expedients applied more or less forcefully. It is not about simply seeing one's own nature, but carving oneself.
In addition, where Zen generally focuses on the effortless aspect of the mind of a buddha, Tibetan traditions focus on the omniscient aspect of the mind of a buddha. Omniscience is often not posited, and sometimes not accepted, by Zen masters. Sam van Schaik wrote a valuable book on the topic Tibetan Zen: Discovering a Lost Tradition.
The Huayan tradition also focused on omniscience rather than naturalness; on expedient means rather than sudden teachings.
The Pure Land tradition was far away from spontaneity, since they though the practitioner was unable to reach enlightenment by its own power. In
Traditions of Meditation in Chinese Buddhism, there is a chapter on the topic. It is called From Dispute to Dual Cultivation: Pure Land Responses to Chan Critics by David W. Chappell.