Let me see if I become clearer.
There is no school in the Chinese Buddhist tradition that is really madhyamaka. Even the Sanlun school, which is said to be its Chinese representative, is not purely madhyamaka.
Tiantai and Huayen are schools that depart from the Madhyamaka nomenclature, but they are not madhyamikas.
Orthodox Madhyamaka does not allow any claims to be made in terms of absolute truth, not even sensitive Buddhist topics such as the Four Noble Truths can be presented as absolute Truths.
However, both Tiantai and Huayen are schools that produce statements. Zhiyi developed a sophisticated scheme that asserts a "nature" of reality, Ichinen Sanzen, just as Huayen developed a similar idea using another nomenclature.
Both traditions rely heavily on texts that are not exponents of the Madhyamaka view. In Tiantai's classification, the Sutra that occupies the apex of the teaching is the Lotus Sutra, accompanied by the Nirvana Sutra. And second is the Avatamsaka Sutra.
Sutra, Nirvana and Avatamsaka Sutras are texts that do not belong to the immediate category of Madhyamaka's primary sources, but rather to the Tathagatagarbha movement, which are the sources that served for the development of Yogacara thought.
Yogacara is not an absolute antithesis to Madhyamaka, but a movement that sought a solution to several problems in the approach of the madhyamikas, problems that naturally end up refuting Buddhism.
The original madhyamaka notion presupposes only two levels of truths: the conventional, or apparent, truth, which is that of existence, and the absolute truth, which is Shunyata, Empytness. However, this statement is illogical, because if everything is without qualities, shunyata, there would be no way to distinguish objects and there would be the process that we call knowledge.
Asanga's job was to try to find a valid answer to understand how Emptyness and Existence can be simultaneous truths, as presented in Madhyamaka. The method for this was based on the meditative practice itself, observing the mind and reality as it is. That's why the school is called Yogacara (Practice of Discipline).
Yogacara is a better developed version of Madhyamaka, and from it several important and crucial elements of Buddhism developed, for example: Eight (or Nine) Consciousnesses, Suchness, understanding of Tathagatagarbha as a positive expression of Shunya, Three Bodies of Buddha, etc. .
No Buddhist school of Chinese origin, not even Sanron, presents their positions without using these concepts developed by the Yogacaras. Sanron's most important text, for example, is not one of the Three Treatises (which gives the school its name), but the Daichidiron, which is possibly an apocryphal of Kumarajiva, and which presents a system that merged the same Madhyamakas and Yogacara conceptions.
It was the Yogacara conception that saved Buddhist discourse and the possibility of presenting and elaborating doctrines, and all Chinese (and most Tibetan) schools are extremely dependent on it. In the orthodox Tiantai lineage of Shakyamuni Buddha, both Nagarjuna and Asanga, as well as Vasubhandu, appear.
The only school that expounds a purely madhymaka position in Mahayana Buddhism today is the Gelugpa, and because it is the school of H.H. Dalai Lama, many Buddhists end up forcing themselves to adapt their school to a strict view of madhyamaka, forcing a supposed general Buddhist "orthodoxy".
In summary, the formulation of Mahayana Buddhism in philosophical terms occurs first in the refutation of the Sarvastivadas propositions (everything exists) by Nagarjuna, which ends up giving rise to the Madhymaka, and later in the rectification of a series of new problems that the madhyamika "argumentation" generates, giving rise to Yogacara, which culminates in the formulation of the general Mahayana thought that is the basis of Sino-Japanese Buddhism.
Sanlun, Tiantai and Huayen represent the first of the systematized schools that emerged shortly after this intense period of debate between conservative Madhyamikas and Reformed Yogacara, and none of them should be considered orthodoxly Madhyamaka.
----------------------------------
As for their existence today, these classical schools still exist in Japan. Todaiji in Nara remains a temple in the Kegon/Huayen tradition. However, the historical formulation of these schools, in China and Japan, did not take place in a sectarian way (the characteristic of Japanese and Tibetan Buddhism today), in which there are schools with a petrified and dogmatic doctrine that needs to be defended by a group of believers.
Sanlun, Tiantai, Huayen, Faxiang, etc. were more study groups and practice coordinators than necessarily dogmatic Buddhist sects. This crystallization will only begin to occur in Chinese Buddhism around the 8th and 9th centuries, when, for example, there is a clash between "internals" and "externals" within the Tiantai tradition in China (Saicho is from that time, studied under a "external" Tiantai teacher, and represents this Tiantai understanding).
This fraternal and open tendency continued to exist in Japan among the oldest temples and still remains a characteristic of Chinese Buddhism. If you ask a Chinese Buddhist which school he belongs to, he will not understand the question, as that is not the heart of Buddhism. The disputes that have existed are related to these attempts to petrify doctrines, disregarding the entire Buddhist corpus.
When Faxiang, for example, crystallized the notion that there are five qualities of beings and five qualities of fruits related to each one, Tiantai said "wait a minute, look, you are saying something from a limited vision, there is this text and this explanation that demonstrates that this is wrong and that it represents only an upaya". But this did not mean abandoning all the formulation and work of other previous, contemporary and later masters.
