Has there been a study or documentary on the influence of Ani-song themes on Japan's music scene?
Unfortunately, I couldn't find a documentary specifically on the subject you requested, purely because I do not think that one currently exists. I did, however, find a number of articles that conducted largely unofficial research on such a topic. In general, has anime music affected the Japanese music industry?

Yes.

How?

I think the most significant effect is the broader audience. Due to anime's rapidly growing international fanbase, and with more and more shows being dubbed in languages other than Japanese and English, the music of anime is being broadcast across the globe. Anime allows the artists of this music to take their songs to the international stage. I think the effect of this globalization if you will can be see across many different anime. 

For example, consider the music artist Linked Horizon. Before 2013, Linked Horizon produced a number of smaller projects, including a song for a Nintendo 3DS game. However, it wasn't until 2013, when the band composed the opening to Attack on Titan, that they truly took off in popularity. Their song rocketed to the top of the leaderboards and was and is a fan favorite across the world. This would not have been possible without anime. 

In addition, there are examples of people changing their whole music tastes and playlists due to their newly found interest in Japanese music from watching anime. You can read here about anime watchers in the US and their changing tastes in music.

It conclusion, the largest effect anime has had on the Japanese music industry is its internationalization. Anime has helped to significantly broaden the audience of Japanese music.

Here's a very good article about anime's affect on Japanese music.