Huifeng wrote:But do these aspects start from Tiantai / Tendai itself, or are they also in turn found in the sources which Tiantai / Tendai used? ie. Tiantai / Tendai is simply continuing the single-practice approach(es) of earlier tradition(s).
Nichiren practitioner here. To the extent I can contribute to the conversation...
How single practice traditions emerged out of the Tendai is a huge subject. You really have to go back and follow the story from China, and from there follow it back into India and the emergence of Mahayana devotional traditions (which will take you further back, and back and back). But to keep this simple we have to set an arbitrary starting point, and the most convenient might be Zhiyi.
IIRC, Zhiyi taught Nembutsu as an aspect of Walking Samadhi in MohoChihkuan. Simply, this is the practice of circumambulating an Amida image while chanting the Nembutsu - a practice that is still very popular at Hieizan, as I understand. Zhiyi supposedly wrote a text on PureLand faith, but scholars doubt its authenticity, and based on what I've read of Zhiyi (in English translation so FWIW) it doesn't sound like Zhiyi.
Parallel to all this, we have the Pure Land traditions. So, I don't know how familiar people are with the Pure Land Sutras - but basically, Amitabha/Amitayus Buddha made a vow when he was still a bodhisattva that if after he became a Buddha, anyone, anywhere, called on him, they would, on their death, be reborn in a lotus calyx in Sukhavati, his Pure Land in the West. In this Pure Land, they would be guaranteed enlightenment. Pureland practice flows from this vow. I'm simplifying it, and some Pure Land traditions have very well developed and subtle philosophies, but that is the nutshell version.
Many Pure Land traditions, however, are very simple and advocate nearly single-minded devotional practice. This tradition has precedent in China (Shantao, for instance), as well as Japan long before the Kamakura period when the other, more well know Single Practice traditions emerged from Tendai - but I'm skipping ahead. These schools of thought became integrated into both Tendai and Shingon traditions by the Kamakura period.
I'll focus on Nichiren, because that's who I am most familiar with. There are other exclusivist schools - like Jodo Shinshu started by Shinran, and Dogen with Soto Zen, but I don't know as much about their emergence.
In the late Heian period, Japanese Buddhists generally concluded that Shakyamuni's teachings had entered the final age of degeneracy and no longer could save people. By their calculations, this age started around 1054. This cynical view engendered what seems like a spiritual crisis - people came to believe that all the tragedies - social and natural - were somehow related to this end of the Buddhadharma. In this climate, Pure Land practice really took off and gained wide popularity - basically, if enlightenment was impossible here, then one ought to pray to Amida to be born in his Western Paradise where Buddhadharma was still vital. The exclusivist Pure Land gauntlet was finally thrown down by Honen who came out and explicitly said - Shakyamuni Buddha's teachings are dead. All practices based on his teachings are futile. There is nothing to do but seek rebirth in Sukhavati.
Basically, in my reading, Nichiren was fighting against the apocalyptic cynicism of Mappo, particularly as taught by Honen. He looked to the Perfect Enlightenment of Tientai/Tendai to redeem the efficacy of Shakyamuni's teachings in this world. At the same time, he seized on the ease and accessibility of Nembutsu practice to make the Perfect Enlightenment possible for all people, not just people with the leisure to sit on a mountain and pursue the catholic approach of Tendai. (Nichiren also thought that authentic Tendai was lost when Saicho died, and really was destroyed when Enin and Enchin took a hard turn toward Shingon-esque doctrines - but that's sort of another subject) To make the perfect enlightenment accessible to all, he boiled the practice down to the Daimoku or recitation of the title of the Lotus Sutra. This builds on Tientai thought concerning the identity of entering the path and perfecting the path. One way to understand is Zhiyi's Six Identities - with Buddha in Principle at the bottom and full blown enlightenment at the top - everyone is on the spectrum of perfection of Buddhahood. Being a novice is, in the final analysis, no different than Buddha. This later developed into Original Enlightenment thought in Japan, but at Nichiren's time, this teaching had not really emerged distinctly yet.
To really talk about Nichiren's theory, you have to have a grasp of Zhiyi's thought. In a sense, Nichiren took the implications of FahuaHsuan-i, ie. that the entire gist of the Lotus Sutra, the teaching of the Eternal Shakyamuni Buddha revealed in the 16th chapter of the Lotus Sutra, is contained in the title of the Sutra. By chanting this title, according to Nichiren, applying the theory of Zhiyi, one thereby undertakes all practices contained in the Lotus Sutra. This in turn is all the practices that the Eternal Shakyamuni undertook himself and taught to others. When the Trace is Opened to Reveal the Root, all practices are revealed to be the practice of the Lotus Sutra. Its the same theory that underlies Tendai catholicism. All life is the practice of the Lotus Sutra. There is nothing that is not the practice of the Lotus Sutra. Chanting the Daimoku is an exclusive affirmation of this teaching, as well as its practice.
I'll leave that there, because going further gets off topic about how Nichiren's single practice emerged from Tendai.