Tendai Buddhism traces its history to the Tian Tai school of China. Tian Tai doctrine, as presented in the teachings of the great master Chih-i, was the first properly Chinese intervention into the Buddhist tradition. This was a matter of necessity. Before and while the Buddhist cultural matrix collapsed in India, Buddhist texts and teachers streamed into China and the Himalaya. There was a kind of chaos: an intense diversity of positions, papers, preachings, practices abounded, and it was not easy for the Chinese to make sense of it all in the absence of a longstanding indigenous Buddhist tradition such as the one that had prevailed in India.
Chih-i offered a systematic explanation for all the Buddhist texts he knew of. Specifically, he organized the Buddhist teachings he knew of in China according to an imputed pedagogy on the part of Shakyamuni: first teaching the Avatamsaka Sutra, but finding it too profound for the capacity of the students around him at the time; then advancing a series of provisional teachings intended to develop those students into fighting shape and prepare them to understand the full and authoritative teachings; finally, presenting the Lotus Sutra, which in Chih-i's mind offered the complete and absolute teaching of the Buddha. (This is a very simplified summary.)
From there, Chih-i explicated the Lotus Sutra in particular ways. I will come back to this matter in the future; for now, what is important to understand is Chih-i's insistence on an all-encompassing, encyclopedic understanding of the Buddhist cultural life, taking it all in and honoring it as appropriate to particular situations. This reflects the Lotus Sutra method of upaya, or skillful means.
This is how Chih-i is reported to have taught just this point:
Someone asked: "In [applying] the true discernment of the Middle Way, through the very act of unifying one's mind, [both] the practice and its function are complete. What need is there to make so many [distinctions] such as the four kinds of samadhi, the application of [discernment] to good and evil [circumstances], or [meditating] amidst the twelve [different forms of] activity? When the water is muddy, the pearl is concealed. When the wind blows heavily, waves beat on the surface. What use could [such concerns] have for realizing lucidity and calm?" Chih-i replied: "[Your attitude] is analogous to a poor person who, upon obtaining a little advantage, considers it sufficient and doesn't care to do even better. If you use only one form of mental discernment, what happens when you are confronted with all sorts of [different] mental states? In such a case you will be at a loss in your own practice. If you [consider] trying to use [this one method] to train others, the spiritual endowments of others are all different from one another. One person's afflictions are in themselves infinite, how much more so [the afflictions of] many people! [Let us say] there is a master of medicines who gathers all varieties of medicines to remove all the different types of illness. Then a person [comes along] who suffers from one particular illness and needs one particular medicine to cure that illness and things it strange that the doctor should carry so many other [useless] medicines [apart from the particular one he requires]. Your question is like this."
(quoted in Daniel B Stevenson, "Samadhi in Early T'ian-T'ai Buddhism," Traditions of Meditation in Chinese Buddhism, pages 85-86.)
In short, one of the reasons Tendai today looks the way it does has to do with an accumulation of means to help people with different needs, like a hospital with a cardiac wing, a renal ward, a trauma unit, and a cafeteria to heal the hungry. We spend significant time working on method and doctrine and debate in order to prepare ourselves for intelligent practice.
One may object that just sitting and not thinking is a legitimate Buddhist meditational practice. Point taken. Sitting there like an idiot is one respectable practice, one of many, but one need not be an idiot about it, nor assume that just sitting is the right medicine for everyone you meet or even for yourself at all times.
One may object that certain Buddhist practices more closely resemble cultural hanging-on from Tibet or Japan or Korea rather than authentic Buddhist teachings, where "authentic Buddhist" means "directly from the mouth of the historical Buddha Shakyamuni." This is a form of back-to-the-text, back-to-the-source Protestantism. I was not present when Shakyamuni taught the Lotus Sutra, or any of the other teachings; historians debate the provenance of this or that document. The broader question has to do with how a particular practice is integrated with a Buddhist view, how it works vis a vis Buddhist view, and how it can help bring some relief to a troubled situation. Even if you do not see how a practice such as nembutsu can be useful to you in your situation, you have no way of knowing if it will be helpful to you or not. Your teacher may have some insight into this, so you might as well ask for guidance from a doctor with some experience healing sick individuals such as yourself.