I find this makes sense, but certain contexts make it seem somewhat different. One classic example are the suttas on the pot of oil and the six animals.
At SN 47.20 (SA 623), mindfulness of the body (kāyagatāsati) is given with a simile of walking carefully with a bowl of oil on one’s head, not looking at a busy crowd with a dancing woman, lest someone cuts off their head. The Chinese parallel interestingly has a somewhat more nuanced phrase, and it defines this practice of mindfulness of the body as all four abodes of mindfulness.
At SN 35.247, we see:
Take a mendicant who sees a sight with their eyes. If it’s pleasant they hold on to it, but if it’s unpleasant they dislike it. They live with mindfulness of the body unestablished and their heart restricted.
…
In the same way, when a mendicant has not developed or cultivated mindfulness of the body, their eye pulls towards pleasant sights, but is put off by unpleasant sights. Their ear … nose … tongue … body … mind pulls towards pleasant ideas, but is put off by unpleasant ideas. …
Take a mendicant who sees a sight with their eyes. If it’s pleasant they don’t hold on to it, and if it’s unpleasant they don’t dislike it. They live with mindfulness of the body established and a limitless heart.
It looks like there are 3 parallels listed on SC: SA 1170, SA 1171, EA 38.8. Here, ‘mindfulness of the body’ is what ties down the senses and allows the mind to be unoccupied by various contacts. It is liked to a ‘firm pillar’. Eventually, the senses (likened to six animals tugging at a rope) lie down beside the pillar.
‘A strong post or pillar’ is a term for mindfulness of the body.
At SN 35.245 and SA 1175, there is a simile of a city with six roads going in and a gatekeeper guarding what comes in and goes out. The city is said to be the body (kāya), the six roads the six senses, and the gatekeeper mindfulness.
There’s also the fact that the suttas talk about ‘asubhabhāvanā’ and ‘asubhasaññā’ rather frequently as a specific practice, and elsewhere ‘kāyagatāsati.’ They could refer to more or less the same thing, but it seems to be referring either to something else or something broader. The Sarvāstivādin Mindfulness of the Body Sutta MA 81 (parallel at MN 119) seems to perhaps be closer to an older original compilation, even though it is longer, as pointed out by Tse Fu Kuan. It seems that the Theravādins may have subtracted the exercises from the original source texts that do not have to do with the ‘kāya’ itself, identical with kāyānupassanā (as you mentioned). The jhāna similes are left because they mention the ‘kāya.’ But the other exercises (perception of light, reviewing sign, cutting off thinking) are from the same source texts for this compilation, so it seems less likely that these harder readings (not about the kāya per se) would just happen to be added later from the same source material in the Sarvāstivādin recension.
This leads some to think that ‘kāya’ here must refer to the six-sense experiencer, rather than the physical body. Bhikhu Anālayo interprets the above practice related to sense restraint as being a form of whole-body proprioceptive awareness where one monitors the sense impingements coming in to the body with mindfulness and a degree of separation to have a broad mind.
One counter example, potentially, is AN 9.11 (parallels: EA 37.6; MA 24)
Sir, someone who had not established mindfulness of the body might well attack one of their spiritual companions and leave without saying sorry. Suppose they were to toss both clean and unclean things on the earth, like feces, urine, spit, pus, and blood. The earth isn’t horrified, repelled, and disgusted because of this.
There is a list of examples given in this sutta, and several of them talk about the body being repulsive, mortal, similar to elements, etc. And it uses the phrase in a way similar to the above examples. So here one gets the impression that it may be about having contemplated the mortality and materiality of the physical body to a degree of disenchantment and detachment.
There is an interesting related phrase, which seems to be left completely unexplained in the early texts:
Kāyagatāsati sātasahagatā. — Mindfulness of the body that is full of pleasure.
(DN 34; SN 16.11; SA 1144; SA2 119: DA 10)
It occurs first and foremost in the Kassapa Saṁyutta (SN 16), as an instruction the Buddha is reported to have given to Mahākassapa. Then it is listed in the DN 34 compilation. I wonder, do you know how the Northern Abhidharma texts or commentaries define and explain this term? It seems at DA 10, the phrase is “謂常自念身” which does not mention pleasure; you translate it as “constant mindfulness of oneself.” There is a difference in the Sarvāstivādin SA version, which mentions “常念其身” but it gives it in a longer context:
『我當正觀五陰生滅,六觸入處集起、滅沒,於四念處正念樂住,修七覺分、八解脫身作證,常念其身,未甞斷絕,離無慚愧,於大師所及大德梵行常住慚愧。』」如是應當學!
There is no translation I know of above. It seems to be talking about the eight liberations (of the “body”-witness (kāyasakkī I assume is the Pāḷi equivalent, for someone who personally attains the eight liberations) and other topics. It does have “於四念處正念樂住” which seems to be referring to joy/happiness parallel to Pāḷi ‘sātasahagatā,’ but here again it references all four abodes of mindfulness for what Pāḷi knows as ‘kāyagatāsati.’ This fits the overall theme I see where ‘kāyagatāsati’ referring to the six-sense person encompasses all four abodes of mindfulness, as Kuan proposed.
SA2 119 seems about the same, but the phrase on the body is different:
宜應專意觀五受陰增長損減,常應觀彼六入生滅,安心住於四念處中,修七覺意,轉令增廣,證八解脫,繫念隨身,未曾放捨,增長慚愧。
“繫念隨身” seems closest to the Pāli wording ‘kāyagatāsati.’
One potential ‘clue’ for the term is a recurring phrase at MA 81, which seems to be explaining how each exercise listed fulfills mindfulness of the body:
如是比丘隨其身行便知上如眞。
It seems this would be implying that it is ‘following the body’s action/activity’ with understanding that is crucial to mindfulness of the body in this discourse.
There’s just some lack of clarity around the term, IMO, in the early texts. But it may be just as you say, mortality reflections and awareness of the body from the perspective of detachment and disenchantment.