Learning how to speak isn't just about accent, although I think having a good (not necessarily perfect) accent is part of the claim to fluency, especially in a pitch-accent language like Japanese.
Some people can get to a convincing accent in a foreign language with very little speaking practice, as long as they do a fair bit of listening. Not perfect, but say 90% there. To give you an example, I learned French in school and had very little practice speaking, and then later went to the same hairdresser in Montreal for four years, and he only realized that I wasn't from France when I told him. Actually practically nobody in Quebec ever seemed to say anything else, and you know that people there are honest to the point of rudeness (Alexandre, I don't know if you're Quebecois, otherwise I'd say you're a case in point

). I don't think they were just trying to flatter me, that would be too anglo. On the other hand, I'm evidently only 90% there because my French (i.e. not Quebecois) landlord heard my non-native accent right away (I wish the same had happened with Japanese, but it didn't).
Another example: in my first year in graduate school there was this guy who my English classmates thought was from Bristol, until he at some point said he was from Brussels and had gone on exchange to Bristol in high school.
So I wouldn't dismiss Realism's claim to having virtually no accent. I would say that if your native language is German or Italian, perhaps also Swedish, it's entirely possible that you have very little accent in Japanese even without much practice. But I have never, ever met a native speaker of English or French who didn't have a noticeable accent in Japanese -- that is, noticeable even to the non-native like me.
But I agree that obtaining a "perfect" accent is entirely possible, but indeed usually requires professional instruction.
In Montreal, for example, there used to be many call centres, and several companies offered training to rid people of their non-native accents (as in India, incidentally). Many French Canadian actors also take accent reduction/elimination training to learn how to speak "standard American" so they can get jobs as extras in US movies. Actors learn to erase their native accents and acquire new accents all the time, so it's definitely not out of the question. Plus you never know how many non-native speakers with perfect accents you've met in your life... the nature of it is that you can't tell.
That aside, I don't think shadowing will do much for you except get the flow of the language right. You're probably better off repeating the native speaker audio recording until you memorize the phrase, so you can use it and modify parts of it when the need arises.