Why?
"Why" is seldom a useful question to ask when it comes to language and words -- it is awfully broad. :)
To narrow the scope a bit, and to focus on the etymology that you mention, let's look at the structure and derivation, and whether these words are related at all.
(Note: This gets long.)
Analysis tips
Underlying word structure
You describe "three-kana kun'yomi". It's important to recognize that spelling is often not a useful lens when looking at Japanese derivations, particularly of kun'yomi, as kanji spellings obscure the structure of the underlying word. For instance, all of your example terms are more than three morae (a "mora" is one "beat" in the rhythm of the word; if spelled entirely in kana, one mora = one kana).
- [謝る]{あやまる} = 4 morae
- [扱う]{あつかう} = 4 morae
- [現れる]{あらわれる} = 5 morae
- [表す]{あらわす} = 4 morae
- [諦める]{あきらめる} = 5 morae
- [争う]{あらそう} = 4 morae
- [改める]{あらためる} = 5 morae
Divining the derivation: finding older forms
Here's your list of seven sample words. I'm including the historical kana, since those older spellings reveal somethings, as well as my interpretation of the underlying forms.
| Modern kana |
Historical kana |
Underlying forms |
| あつかう |
あつかふ |
あつく* + ふ |
| あらわれる |
あらはれる |
ある + ふ + れる |
| あらわす |
あらはす |
ある + ふ + す |
| あらそう |
あらそふ |
あら + す* + ふ |
| あらためる |
あらためる |
あらた + む |
| あきらめる |
あきらめる |
あきら + む |
| あやまる |
あやまる |
あや + む |
Divining the derivation: common suffixes
We see a number of common endings combined in different ways: ～ふ, ～す, and む, and one ～れる. Let's look at these in turn.
～ふ
This is a [助動詞]{じょどうし}, often glossed as "auxiliary verb". These were very productive in Old Japanese, and were used as suffixes to derive new senses from existing verb roots. The auxiliary ～ふ was used to indicate a repeated or ongoing action or state. Here's the Kotobank page with entries from various monolingual Japanese dictionaries.
The ～ふ auxiliary attached to the [未然形]{みぜんけい} or "irrealis form" (where "irrealis" = "hasn't happened yet" or "hasn't completed yet"; see also Wikipedia) of a verb. This fits for most of the cases above where we have ～ふ.
As a regular sound shift, medial (mid-word) //f// sounds (technically the bilabial consonant [[ɸ]], not the labiodental consonant [[f]] that we have in English) shifted to //w// before //-a//, and generally vanished otherwise. This is why we have things like [買う]{かう}, but also [買わない]{かわない} with that //w// that appears -- this comes from older //kafu//. So the //f// in ふ shifted and disappeared, so all the verbs that ended in ～ふ now end in ～う instead.
⇒ If you see a modern verb that ends in ～う, it might be from this iterative / repetitive / stative auxiliary ～ふ. Consider the meaning, and whether the part before the ～ふ might be another verb in the [未然形]{みぜんけい} conjugation ending in //-a//.
～す
This is ultimately cognate with the everyday verb する ("to do"), and the causative endings ～させる・～せる. It serves to make a verb causative ("make something do") or transitive ("do something to something else").
～む
This appears to be ultimately cognate with the everyday verb [見る]{みる} ("to see") and noun [目]{め} ("eye"), and probably also the suppositional / volitional auxiliary ～む that became the modern -ō or -yō ending, as in [書こう]{かこう} or [食べよう]{たべよう}. The basic meaning in your sample verbs above is something like "seems like, having that quality" in the intransitive form ～まる, and "make something seem like, make something have that quality" in the transitive form ～める.
～れる
This is from older ～る, possibly in turn cognate with ある ("is, to be"). The basic sense is "happening spontaneously, or caused to happen by something else", basically in line with the modern -areru or -rareru passive / potential ending.
Divining the derivation: each word
Let's look at each of the words in your sample list.
あつかう from あつく* + ふ
This is described on Kotobank in the Kokugo Dai Jiten (KDJ) entry as derived from [熱い]{あつい}・[暑い]{あつい} ("hot"), from the idea of worrying oneself about something. Consider the English expression "hot under the collar about something". Over time, this shifted to "worry oneself about something and get that something done", and from there to "getting something done", as in "handling something". The ～ふ appears to imply ongoing or repeated action.
あらわれる from ある + ふ + れる
This appears to be from ある ("to come into being") + ～ふ ("continuing or ongoing state") + ～れる ("spontaneous action happening on its own"). Thus, "to be coming into being on its own" → "to appear, to manifest".
あらわす from ある + ふ + す
This is the causative / transitive counterpart to あらわれる, and appears to be from ある ("to come into being") + ～ふ ("continuing or ongoing state") + ～す ("causative / transitive suffix"). Thus, "to make something come into being" → "to make something apparent, to make something manifest" → "to reveal, to show, to expose".
あらそう from あら + す* + ふ
This appears to be ある ("to come into being") → あら (irrealis form, "not done coming into being", i.e. "raw, rough, unrefined" → "rough") + ～す ("causative or transitive suffix") + ～ふ ("continuing or ongoing state").
The ～す* here is more speculative. The expected form for ～ふ to attach would be あらさふ, but the oldest texts (even as far back as 712) record this term as あらそふ. There is a sound shift from //sau// to //sou// in Japanese, but that happens much later. I have also noticed an apparently //-a-// ↔ //-o-// alternation in various ancient vocabulary terms, where the //-a-// variant has nuances of "external, outward, surface" and the //-o-// variant has nuances of "internal, inward, inherent", and I wonder if this might be one such example.
We can clearly point to almost-synonym あらがう, as referenced in the [語誌]{ごし} section of the KDJ entry here, pointing to the root あら～ as the start of the verb.
あらためる from あらた + む
This is [新]{あらた} ("new") + ～む ("to seem like, to have that quality") in the transitive ～める form ("to make something seem like, to make something have that quality"). Thus, "to make something seem new, to make something have the quality of new-ness" → "to renew", and by extension, "to improve something; to correct something".
あきらめる from あきら + む
This appears to be [明]{あきら} ("bright, clear, distinct, visible") + ～む ("to seem like, to have that quality") in the transitive ～める form ("to make something seem like, to make something have that quality"). Thus, the original meaning may have been "to make something clearly visible", as described in the Dajisen entry here at Kotobank. I'm not sure how the modern "to quit" sense might have arisen; perhaps from "make something clear" → "make one's intentions clear (to stop doing something)" → "to quit"?
あやまる from あや + む
This appears to be root あや as in [危うい]{あやうい} ("risky, dangerous") or [危ぶむ]{あやぶむ} ("to worry that something is risky") or [怪しい]{あやしい} ("odd, troublesome, vexing, irksome") + ～む ("to seem like, to have that quality") in the intransitive form ～まる. According to the [語誌]{ごし} notes in the KDJ entry here at Kotobank, the original sense was something like "to draw close to peril", shifting from there to a transitive sense of "to acknowledge one's mistake (in drawing close to peril)" → "to apologize".
Conclusion: not all related, but some are
There is only a limited set of roots possible that start with あ, and these roots appear in various derived terms that have developed over the years.
Please comment if the above does not address your question, and I can edit to update.
