The popular idea that "arigato" derives from "obrigado" has been repeatedly debunked. However, every debunking I have seen is unsatisfactory. It is pointed out that the Japanese word from which "arigato" is derived long predates the Portuguese arrival in Japan. And that's it. But that's clearly unsatisfactory. A satisfactory debunking would claim, and demonstrate, that the word itself, in that specific use, predated the Portuguese arrival. Without this, it remains possible that the following happened:
1) "arigatai" meaning "difficult" long predated the Portuguese arrival, but the use of "arigato" to mean "thank you" did not predate the Portuguese arrival.
2) Observation of the Portuguese saying "obrigado" to mean "thank you" caused the Japanese to say "arigato" to mean "thank you".
I do not have specific examples, but I do believe I have seen genuine examples of this sort of thing happening with other words in other languages - that is, that one language has indeed affected another language by altering the second language's use of its own native words.
Nothing I have read anywhere specifically excludes this possibility. Talk is always about the word "arigatai" meaning "difficult" predating the Portuguese. In my experience (and I have looked into this multiple times over the years) no one ever offers any samples of Japanese writing "arigato" to mean "thank you", or even claims that they did so before the Portuguese arrived.
So, no one seems to have ever debunked this possibility. If this is what happening, then it is simply overstating the case to claim that:
"Superficial appearances notwithstanding, there is absolutely no linguistic relationship to the Portuguese word obrigado of the same meaning."
If the Portuguese use of "obrigado" shaped the Japanese use of "arigato" then that is a linguistic relationship.
I would be happy if either:
1) Someone finally specifically showed that "arigato" was used to mean "thank you" before the Portuguese arrived, or
2) People stopped overstating the case against the relationship between "obrigado" and "arigato".