Azusa, I am curious, why would you assume that learners who pronounce the sounds right will get the pitch wrong? I would assume that learners get the odd sound wrong and the odd pitch wrong.
Alexandre, I will assume that you know much more about pitch than I do, and that you are right about my wrong pitch in the recording. I have no idea if I get the pitch wrong all of the time, most of the time or just some of the time. Whether I get some words wrong all of the time, or just some of the time.
I am not at all motivated to deliberately work on pitch. I can imagine that others might be, but to me it is a distraction, and not a form language learning that I would enjoy. That is just me.
If I wanted to improve my Japanese, without being in Japan, I would read more, and listen to audio books. It is quite possible that if I spent more time listening to Japanese, my pitch would naturally improve, without thinking about it. But it is not something I would want to focus on.
When I hear a non-native speaker of English who uses the language well, I am impressed and would not be put off by an accent, and would not dream of correcting the accent.
The pursuit of perfect pitch may be of interest to some, but I do not think it is important, and that is my major area of disagreement with you.
Alexandre, you ask me how I know that my accent, or that of certain other non-natives, is good, certainly good enough. The answer is that I see them operating very effectively and I know that I have no trouble communicating in all kinds of situations in Japanese. That is good enough. It is good enough for my Japanse, and good enough for most of the non-native English speakers who master English.