I suppose one could study any style of Martial Art, and proceed on, without any regard for it's history; and remain happy and content.
Others may have their interest "perked", and review some brief history. They may, perhaps, discover conflicting accounts. They may also assume, and presume into error, as far as the historical facts that can be gotten are concerned.
Often times an Art's history becomes shaded by biased. It's almost natural for this to happen. In many cases, one's emphasis may be dependent upon perspective. Conversely, ones perspective may be greatly influenced by emphasis.
I happened upon this statement, from a reliable Chinese source. It comes from a purely Chinese perspective. It's emphasis is upon the history of Chinese Tiger Boxing.
"Li Zhao Bei however traveled to Fuzhou and opened a school there where he accepted disciples such as Yang San Shan (est.1810-1873) and others. Zheng Bu Su (also known as Zheng Xianji (郑仙纪, 1854-1929)) was one of the most famous masters who opened a school in Fuzhou and had taught many disciples, the most well known was Zhou Zi He (周子和, one of the most famous fighters in all of Fuzhou’s history). Zhou Zi He mastered many systems of Fujianese martial arts but his most favored was that of Tiger Style. In fact, Zhou accepted a Japanese student who studied some basics and went on to be the founder of Uechi Ryu style Karate".
The source in it's full context can be referenced here - http://www.satirio.com/ma/huzun/history.html
Perhaps some have "overemphasized" Shu Shiwa's influence in Uechi Ryu History. There are at least two other sources that should be taken into consideration, when speculating upon the influences that spawned Uechi Ryu. The first would be the Ryukuko where Kanbun first trained in China. Though he ended his schooling at that location, there is no reason to think that he completely discarded what he learned while training there.
Fuchow, in Fujian provence, was somewhat of a Chinese boxing "Mecca" at the time. No doubt exposure to many influences would have been a consistent experience to those dwelling there.
Verbal tradition also says, that Kanbun Uechi was instructed in some form of Chuan Fa, within the precincts of a Temple. I looked into the probability of such an account: and, even a cursory investigation will reveal that at least FIVE Temples; were, opened and functioning, within the confines of Fuchow City, at the time of Kanbun's soujourn there.
There were many more in close proximity to the location, in the environs of Fujian Provence. Uechi Kanbun was very proud of the fact that some Temple Chuan Fa had been incorporated into his style! And, never rescinded the account.
An early history of Uechi Ryu, claims that Kanbun Uechi Sensei studied the three most prominent styles extant in Fuchow at the time; and from these elements, devised what would become Uechi Ryu.
It's the History that I believe. I think that when displaying the "Lineage" of Uechi Ryu, the first, and only first name, should be Kanbun Uechi Sensei. All else, is "Speculation"!
I believe Uechi Kanbun devised Uechi Ryu. I also think he deserves the credit for it, and should, to some degree, be considered a "Genius"!
In digression, for those who are interested in the history of Chinese Martial Arts, I found this source reputable and captivating. http://www.bgtent.com/naturalcma/index.htm
In closing, when viewing the history of Okinawan karate, one will find that virtually every "Style" was the inspiration of one Okinawan's genius, applied to what he had been taught, and exposed to. Why should Uechi Ryu be any different? After all, the process itself, is nearly an Okinawan Tradition!
All The Best!