Your question has two parts: What did the Buddha mean, and is the phrase "breakup of the body, after death" correctly translated. It seems clear that the latter is correct, and a defense of a positive answer has already been given by Gabe, so I won't address that.
"What did the Buddha mean" can only be a matter of opinion, since none of us is able to directly hear his thinking or ask him, but it is possible to look at the sutta for some answers, and consider the larger context of his talks in other suttas, to try to understand why he says what he says in AN 5.38.
What I notice about AN 5.38 is that
- We aren't told who is speaking, but we might well assume it's the
Buddha.
- The talk is not a deep one as would be given to long-time monastics,
and it is not being given to the clansmen (using Bodhi's translation
of kulaputta).
- Four of the five points address what benefits accrue to the
clansmen.
- The point that doesn't follow the main pattern seems to be
addressing a benefit to the monastics, not to the "clansmen endowed
with faith". It is that the monastics get alms from a person of
faith.
Assuming it's the Buddha speaking, it's worth noting that it's not a deep dharma talk illuminating the finer points of his teaching. It appears to be a talk about why regular folk ("clansmen") who have faith are wise to have faith, and as such it would be addressed to new monastics, not long-time, experienced monks. One needs to question why the Buddha would be instructing inexperienced monks on the reasons the faith of clansmen is beneficial. That one of the five points he lists is of benefit to the monks might then give us a clue -- why should the monastics be kind to faithful clansmen? for alms!
Why then are these monks being told what benefit faith has for the clansmen? What do monastics do to gain alms from clansmen? They offer teachings. What teaching might novice monastics give to clansmen? How about a talk on why their faith is of benefit to them: It's because, dear clansmen, you'll get compassion from others, you'll get such a good reputation people will come to you, you'll get taught dhamma before anyone without faith does, you'll get a good rebirth, and (monastic holds out his bowl), you're known to be generous.
It is worth noting that when the Buddha speaks to individuals he encounters (not speaking of his devoted monastics here), he speaks to them addressing their concerns and without making dogmatic statements about what they should believe. He often enough points out the obvious consequences of what they believe, but he doesn't, for example, go around telling people they are just wrong to think that behaving like a dog will get them reborn as a dog in dog heaven, or even that there is no "self" -- he never says that. But he does point out the ways the behavior their thinking inspires in them leads to trouble, or that there is no discernible self to be found that matches what determined a "self" in their times -- that it was a separate thing, eternal, changeless. What he does do is address an individual's concern from within that individual's own worldview, pointing out any obvious flaws in the consequences of their thinking. He takes pains not to argue. He even says this.
In MN 18.4 he says, "Friend, I assert and proclaim [my teaching] in such a way that one does not quarrel with anyone in the world with its gods, its Maras, and its Brahmas, in this generation with its recluses and brahmins, its princes and its people; in such a way that perceptions no more underlie that brahmin who abides detached from sensual pleasures, without perplexity, shorn of worry, free from craving for any kind of being." SN 22.94 opens with "Bhikkhus, I do not dispute with the world; rather it is the world that disputes with me."
It seems apparent, to me at least, that in this sutta he is giving his monastics directions to go out and do what he does: give a teaching that fits within their worldview as to the reasons faith is worthwhile. Their worldview includes the belief that good behavior in this life leads to a better rebirth. If the world works the way they think it does, then this, too, would be a benefit of their faith. That doesn't mean the Buddha is sure that rebirth is the cosmic order. Even if he did, he would not say so because he'd be quoted on that and that would lead to 'proclaiming his teaching in such a way that he quarrels with the world'.
I do note that there are places in the suttas where he does seem to suggest that he has had direct experiences of his past lives. Why he does that would be part of a much larger, longer discussion. I further note that he sometimes tells people where their loved ones will be reborn after death. He himself answers why he does this, in MN 68.8-9.
"What do you think, Anuruddha? What purpose does the Tathagata see
that when a disciple has died, he declares his reappearance thus:
'So-and-so has reappeared in such-and-such a place...'?"
"Anuruddha, it is not for the purpose of scheming to deceive people or
for the purpose of flattering people or for the purpose of gain,
honour, or renown, or with the thought, 'Let people know me to be
thus,' that when a disciple has died, the Tathagata declares his
reappearance... Rather, it is because there are faithful clansmen
inspired and gladdened by what is lofty, who when they hear that,
direct their minds to such a state, and that leads to their welfare
and happiness for a long time."
