From someone who has watched a lot of anime, I was actually a little confused about what trend you were talking about, up until you mentioned it being maybe a problem with shōnen anime and talking during fight scenes. It's an obvious and much-commented upon quirk of action shows, so I would say that yes, that's a big part of why you're picking up on it, but there are also some other points to be made.
To first talk about shōnen anime, it, like anime generally, takes much of its visual style and representational tropes from manga, which uses a lot of that kind of mid-fight exposition because of the necessity of explaining to the reader the technicalities of the fight. It's a bit like the difference between sports and esports in that regard - physical sports is relatively easy to grasp even for people with a tentative knowledge of the rules because they can see when something is impressive by comparing it to their own physical abilities, wheras with esports the abilities of players are so abstracted that a knowledge of the rules is much more necessary. In manga, the abilities of characters are often completely divorced from reality, so this exposition is used to "explain the rules" to the reader. It also doesn't hurt that retaining this particular quirk means that anime companies can fill in time with a focus on the speaker's/attacker's face, making it partly a budgeting issue. If you compare something like Bleach to the fantastically animated Fate/Zero and recent Fate/Stay Night adaptations, they don't rely on the technique nearly as much.
Outside of shōnen stuff, though, there could be a couple of ways you're picking up on a perceived problem of systematic over-explanation (note I am not saying that Japanese people/media/whatever "never" overexplain things). One is in your differing expectations about noun-verb collocations (that is, what verbs get paired up with what nouns to describe an action). In samurai dramas, for example, characters will often say they will "cut" someone, meaning to kill them with their sword. Depending on how things are translated in the media you're watching, you might get the impression that characters are over-specifying (as an extention of over-explanation) because you're not familiar with what is standard usage of verbs in Japanese.
Another issue related to translation is that English is a prestige language in Japan, meaning that often people (and characters) will throw in English words and expressions to sound fancy - much like some people do for French in English-speaking countries, and especialy Latin in English academia. The primary way that things are translated, though, especially if you are watching things dubbed, is to make the product sound as though it were originally written in the target language, so J-E translations often fumble around instances where an English word is used followed up by the Japanese word (maybe the character they're talking to doesn't understand the English, as an example of one situation), and this too could come off as a problem of overexplaining, where what you think is (through the translation) the "same word" has come up twice in a row.
CONTINUING, something related to this is is a class of word called "aizuchi," which is the verbal feedback you give to a speaker to indicate you're listening, like "yeah," or "uhuh." Japanese etiquette puts a lot of emphasis on these kinds of expressions as a way of communicating, "okay, got it, keep going." So, when you're playing Metal Gear, and Snake grumbles incredulously, "METAL GEAR?" or "The Patriots..?" for the fifth time that minute, it's actually not the game "over-explaining" to Snake because he's an idiot and apparently hearing the information for the first time, like a lot of people who have only played in English believe, but rather a misunderstanding of what is being communicated with those utterances, either by voice actors or the voice director.
I thought I'd also put in a few thoughts on "aimai," since you've brought it up. The phrase means "vagueness," first of all, but is often used in the sense of being "wishy-washy," so it isn't necessarily a good thing. The preference towards vagueness is mostly realised when it comes to emotions, like in romantic relationships, and aesthetics, but doesn't really apply to explanations in sci-fi, or style in something as low-brow as children's combat cartoons. If anyone is trying to tell you that vagueness is an undercurrent of a huge amount of Japanese culture, they're buying into a certain fetishised picture of "wierd" or "unknowable" Japan. I believe that this is perpetuated by people who go to the country barely literate in the language and not having studied the culture in any significant way. After all, any culture has its own verbal and physical cues that are immediately understood without needing further explanation - and modern accounts of Japanese people being unscrutable come down to not being versed in those things that they have picked up over a lifetime. I assure you that Japanese people often feel the same way about (particularly white) non-Japanese.
This is an interesting topic, Sage, so I hope that this giant wall is of some help to you. If you're wondering if you can trust this any more than the word of the people who you bring up in your post, consider that I've studied the language for seven years, do postgraduate research in Japanese-English translation studies, and tutor the Japanese culture classes at my university.
