The plural ending -(r)en
Let’s get rid of the plural suffix first, since it is actually quite unrelated to the main question here—it is merely a red herring.
The plural ending -en is the outcome of the Old English (OE) ending -an, which was the regular ending in the weak noun declensions. As you can see, this ending has an a: as such, it did not cause i-mutation. Of the words mentioned in the question, ox is an old weak noun, and it has maintained its old plural suffix -en as a weak noun, though the class of nouns known as ‘weak’ has long since disappeared.
The other two, children and brethren, are not originally weak nouns, but they have taken over the weak ending later on.
Children, unlike brethren, does not display i-mutation in the plural, nor anywhere else—i is simply the root vowel in that word in all forms. The word was (as the source you linked to says) originally part of a minor declension type called the -r-plurals. These were originally, for the most part, athematic s stems, which accounts for the extra r in the plural: in Proto-Indo-European (PIE), their nominative/accusative singular forms ended in *-os (same as the thematic nouns), while the plural simply added on the athematic plural ending *-es, yielding *-oses. In Proto-Germanic (PG), these became regularly *-az and *-aziz.
In West Germanic, final syllables were lost entirely, so *-az disappeared in the singular, as did *-iz in the plural; but in the plural form, *-az- was not final and therefore only lost its vowel (sometimes not even that): the consonant was ‘protected’ by the final *-iz while it was still there. PG *z intervocallically became *r in most later languages, so *-aziz gives OE -(e)r(e) or -(e)r(a) (final vowels display quite a bit of variation). PG *kilðaz then became OE ċild, while the plural *kilðaziz became ċild(e)r(e) or ċild(e)r(a) quite regularly—an ‘r-declension’ was born.
When that declension was subsequently lost again some time in early Middle English (ME), its members had various more common plural endings added—some to the singular stem without an r (like lambs), some with the r (basically children; can’t think of any others here). Ċild(e)ra then took the originally weak plural ending -en (< OE -an).
Similarly, brethren represents an older form (which did have i-mutation) that had the weak plural suffix added on to it in Middle English (see below).
Brethren: i-mutation
Brethren is really the only odd one out that does have i-mutation, and it is indeed very odd.
The OE plural was identical to the singular, just like with the other kinship terms in -r (fæder, mōdor, sweostor, dohtor). They must be considered an irregular group in OE, as in many other languages: they represent a small group of kinship terms that all contain an old kinship suffix, Proto-Indo-European (PIE) *-ter, and they act as a group in many languages, being often at least somewhat irregular.
The plural found in the other Germanic languages for these words is akin to the one found in the class of athematic nouns in OE: the main plural marker is i-umlaut, from an earlier PG *-iz, from PIE *-es, the regular athematic ending. This is different from the OE plural, which (to my knowledge) has not been satisfactorily explained.
If we look at brother, the most common ending in OE is indeed brōðor, just like the singular; but many variant forms are found, among them bro͞eþre (Mercian), representing /ˈbrœːθrə/, which is basically what we would expect the OE outcome of the form found in the other Germanic languages to be. We have to assume that forms like this, with i-mutation and an athematic ending also existed elsewhere dialectally in OE apart from Mercian, but it is very peculiar that they are so extremely scantly attested.
In the ME period, the old plural of brōðor, identical to the singular, was gradually abandoned in favour of more distinct forms—particularly ones that look like the Mercian one above. Broþre and breþre were among the most common ones, and just like what had happened to ċild(e)r(a), forms with an added-on old weak stem plural -en arose and became very common, too: broþren and breþren.
Eventually, when the plural marker was unified across the entire language in favour of -s, brother was regularlised as brothers; but the form brethren managed to survive in a slightly distinct sense.
As should be clear now, however, the i-mutation in brethren does not stem from the -en suffix (which, as per above, did not cause i-mutation): it was already there before the -en suffix was analogically added to the earlier plural form.
Children: vowel change
As you mention, the singular of child has a long i /tʃaɪld/, while the plural has a short i /tʃɪldɹən/. The reason for this is quite separate from i-mutation or anything of the sort. Unlike i-mutation, which took place in common North-West Germanic, i.e., within the first three or four centuries AD, this is a purely English thing.
In late OE, there was a regular sound law that lengthened any vowel before a homorganic cluster, such as /ld nd rd mb ŋɡ/. This change is often referred to as homorganic lenghtening. It meant that the earlier OE form ċild [cild] or [tʃild] now became [tʃiːld], just like the similarly structured kind [kind] (borrowed from Norse) became [kiːnd]. When the Great Vowel Shift came along some centuries later and generally mucked things up, /iː/ eventually became [aɪ], and [tʃiːld kiːnd] became [tʃaɪld kaɪnd].
A bit later than homorganic lengthening came pre-cluster shortening, whereby a long vowel was shortened before a cluster of three or more consonants. Thus, the plural childre(n) (and also, incidentally, brethre(n)) went from having long vowels [tʃiːldrə(n) breːθrə(n)] to having short vowels [tʃɪldrə(n) breθrə(n)], while the singular child retained its long vowel.
