Exactly my (albeit low educated) thinking, too. If I remember correctly, I was told by someone that there are people who are quite good at internal integration (my words), but cannot apply it well in a combative setting; for them it's a healthful practice...but they can do "it." How can we balance ourselves, our many aspects, to suit the situations that arise, to address the "middle of now?" Application practices.
At the end of the day, internal strength is an unusual kind of strength that confers certain advantages on the practitioner. Depending on the skill of the practitioner and the specific application (lifting, fighting, gymnastics, calligraphy) - there may be other attributes that more than offset any advantage that internal strength can confer. One reason it's not given as much press in the Japanese arts, I'd wager, is two-fold: 1) It's assumed to be one of the skills already embedded into the practice of a particular system 2) It's not understood well enough on it's own merits to be recognized as to the value it brings when trained appropriately and partnered with other attributes in the context of a martial art.
One reason I'm challenged by (1) is that in an art like aikido - it basically assumes that you have some degree of internal strength at work in order to appropriately train the techniques (classic examples being extending your ki/jin past a person's balance point while controlling the connection points between you and another person as if you were one unit). You don't need such a thing to practice the technique, especially in non-competitive "jujutsu" type circumstances and there's enough other things going on (ukemi, ma-ai, tai sabaki) to coordinate all at once that I'm not surprised that "internal strength" principles don't get much more than lip service, especially in larger groups.
I'm pretty convinced that the "shape" of aikido (and arms-length jujutsu) techniques are a pretty ideal construct in which to practice internal strength from a body-to-body limited connection perspective (focus extending ki/jin through specific points) via your connected body into the other person's body by joining your bodies together with connection and intent-based work. As you get better at manipulating the connection in you and another, then the additional aspects of ukemi, ma-i, tai sabaki, etc. can be welcome enrichments to the container that is your martial art. I do think some degree of sparring (body/body grappling, push-hands based on jin/connection rather than awkward wrestling) would be useful as well as some practice in power releases (can be trained well in suburi if you know what you're training) would be excellent secret sauce to one's atemi in a gokui sense.
In more sports-oriented combatives, it's seemingly less important to have internal strength be a part of the equation to have functional skills because you are already adding pressure and testing them earlier on. If you look at the functional end products of internal strength - increased sensitivity and the ability to be stable and release a lot of power in seemingly unusual ways - it might not be as useful in some sports combatives where there are ways that are 1) much faster time to market in terms of usable skill 2) even mid-level internal strength skills may be worked around by someone more adept within the application rule-set at winning their game (e.g. Shaolin monk gets knocked out by trying to absorb too many punches to the head).
So to my first paragraph, regarding partnering internal strength with other attributes (point 2) into the container of a martial art requires some additional challenges and considerations beyond your internal strength skill and conditioning. What's your delivery system (e.g. aikido wrist grabs and overhead strikes) to apply your abilities? How are you measuring your internal strength and ability to apply it? (resisting a push is useful to measure some baseline accomplishment but not very practical in a fight)
My personal suspicion is that aikido as a delivery system was intended to be one of intercepting an oncoming attack through an internal strength powered entry (which in some cases would end an encounter in the sense of an unskilled attack), then if met by a skilled opponent, yielding appropriately only to re-enter again from an advantage (balance control, optimal targeting, power release opportunity), the "aiki" portion assumes your connected body is also in an advantageous position such that any effort they make to apply power to you is joined to your connected power and reapplied back against them.
So the three component pieces in an aikido sense all need to be trained. Internal strength conditioning (body rewires to move in a different way, more sensitive, more cohesive with unusual power), internal strength skill (ability to leverage internal strength advantages "on demand") and the ability to express internal strength principles via the "shape" of something recognizable as aikido. In my world, you can train all three concurrently (they won't actually come together until some discrete time has been focused on internal strength conditioning and skill) but you have to somewhat realize where one begins and the other ends (I think that's why Tohei had rankings explicitly in "ki" versus "aikido").
Anyways, as always, ymmv.