As pointed out in other answers, わかる is etymologically an potential/intransitive/passive verb of わける/わく. Then there are two questions:
- why cannot potential/passive/intransitive verbs have potential form.
- why the intransitive わかる is always used.
- how the potential form works
Why cannot potential verbs have potential forms
I have had this questions for years and my explanation is: making the object the subject is just the way to express “result” in Japanese. A “result” is something you can assert it happens/will happen/happened only after if actually happens. It is not something can be controlled by the agent.
It is common to see people say お茶が入った, 風呂が沸いた, ご飯ができた, etc. Some people think using intransitive verbs are more polite and being polite is a feature of Japanese. However, I hardly think so.
It is interesting that in Chinese, we say neither 1~3a nor 1~3b. Instead, we say something like 1~3c. The point is, the transitive verb and passive verb are used to report the whole process.
1a) お茶が入った
2a) 風呂が沸いた
3a) ご飯ができた
1b) お茶を入れた
2b) 風呂を沸かした
3b) ご飯を作った
1c) 茶沏好了 (お茶が、入れられて、うまくなった)
2c) 水烧开了 (風呂が、沸かされて 、沸いた)
3c) 饭作好了 (ご飯が、作られて、できた)
In a context where some action is supposed to be taken / have been taken, if you just want to report the result, which has no duration, you must use a different expression, that is, the intransitive form.
The difference between Chinese and Japanese is that in Japanese you can just make the original object the subject of verb without explicitly mentioning its cause, while in Chinese the relation between the consequence and its cause must be explicitly conveyed. I am not good at English and I do not know how what expressions will be used in this context in English.
Because the result is not something we can control, we might be interested in what the result is supposed to be, ie its potentiality. When there is an unstated agent and his effort in the context and no adverbs, it can be usually assumed that an adverb うまく (successfully/skillfully) is implied. (Historically, 能く{よく} (well, skillfully), which was similar to うまく, was another way to express potentiality)
4a) ご飯が作れる
5a) 漢字が読める
6a) 運転ができない
They roughly equal their more verbose conditional versions.
4b) ご飯を作りたければ、作れる
5b) 漢字を読めたければ、読める
6b) 運転をしようとしても、できない
In other words, the difference between “will” and “can” mainly depends on the context -- whether there is an implied effort or not.
In the last sentence 6, the distinction between “will” and “can” are neutralized in negative statements, which is unexceptional because historically, the “potential” function of られる only appears in negative sentences.
For example, 得ず{えず} means both “will not get” or “cannot get”, 能わず might be the intransitive verb of あたえる, which means “will not get” or “cannot get”, too.
Why わかる is always intransitive
I do not have a confident answer. But I think, there are several verbs of perception that are usually intransitive, such as 見える, 聞きこる, わかる.
I think it reflects the process of human mental activity. We are always looking, listening, observing the circumstance around us, and thinking about the information we perceived. So what concerns us is not whether we performed the action or not, but the result we got.
I looked around, but we do I see? The result will be 靴が見えた.
I though about it, but we do I realize? The result will be お客さんが来ていることがわかった.
Maybe because their intransitive versions are frequently used, they are irregular. The active wakeru or waku might have never been used to mean " to understand". I think I have read a paper about the etymology of wakaru, but I cannot find it.
The weirdness of the potential form
Although I consider the potential form is as a special case and productive form to express “the result”, it do not mean it always works as you may expect. Compare:
10a) ご飯ができる
10b) ご飯が作れる
10c) ご飯ができない
10d) ご飯が作れない
11a) 富士山が見える
11b) 富士山が見られる
11c) 富士山が見えない
11d) 富士山が見られない
We can see a clear difference in how these action succeed or fail. Some forms, such as ご飯が作れる and ご飯ができない are not likely to be used.
However, adding some adverbs will make a difference.
12a) ご飯がおいしくできる
12b) ご飯がおいしく作れる
12c) ご飯がおいしくできない
12d) ご飯がおいしく作れない
We can even use adjectives.
13a) おいしいご飯ができる
13b) おいしいご飯が作れる
13c) おいしいご飯ができない
13d) おいしいご飯が作れない
It is a little hard to determine the naturalness and differences between all these expressions, but the meaning of potential form is not easy to explain.
With adverbs expressing desirable resultative state, the difference between the potential form and the active/passive form can sometimes be more or less neutralized.
14a) 絵がうまく書いてある
14b) 絵がうまく書かれている
14c) 絵がうまく書けている
15a) 英語を話す
15b) 英語を話せる
15c) 英語を流暢に話す
15d) 英語を流暢に話せる
Since うまく and 流暢 are desirable state, it can usually be safely assumed that they are the results of the agent's effort.
For some verbs, the passive form can be used instead of the (regular) potential form.
17) 残念に思われる
18) 残念に思える
Sometimes, non-volitional verbs can have passive forms
30) 得る <> 得られる (得る has become a volitional verb now)
31) 知る <> 知られる
Some potential/passive forms become volitional verbs and can further be put into the potential form.
32) 痩す <> 痩せる
33) 生まれる <> 生まれられる
34) 覚える <> 覚えられる
For some verbs, they are primarily passive/non-volitional, the potential forms are not often used.
35) 教わる <> ?教われる, but 教わることができる
36) 知る <> ?知れる, but 知ることができる
