I think there is a lexical sensitivity to the choice between 에게 and the more formal and universal 에 의하여 (for 'by').
That is to say, most any active form can go into the passive and take 에 의하여. Your examples would become:
- 수프가 요리사에 의하여 섞이다.
- 문이 남자에 의하여 닫히다.
But only a subset of verbs can take 에게 (or more colloquially 한테) in addition to 에 의하여.
I want to offer these as broad generalizations, but am somewhat dubious about their utility:
Passive verbs taking 에게 are probably those used often.
Certain "에게 passive verb" constructions can only have a human, animate or "personified" subjects.
For item 2, consider:
- 경찰이 범인을 잡다. (The police catch the perpetrator.)
- 군인이 총을 잡다. (A soldier grabs/holds a gun.)
- 늑대가 어린 양을 먹다. (A wolf eats a little lamb.)
- 늑대가 양고기를 먹다. (A wolf eats mutton.)
Note: Except in the police the assignment of a definite or indefinite article was arbitrary.
Only two of these have passive counterparts.
- 범인이 경찰에게 잡히다.
- 어린 양이 늑대에게 먹히다.
The following do not work:
총이 군인에게 잡히다.
양고기가 늑대에게 먹히다.
What this shows is that the shift from the active to the passive verb involves a semantic change. I.e. the passive verb is a word having a narrower range of meaning than the active.
Further subtlety comes out when we consider dropping 에게.
- 손잡이가 (구도가) 잡히다.
The first means something like the handle is within reach or the composition is attainable. 먹히다 works in the sense of appeal, a very informal usage.
내가 애들한테 좀 먹히잖아. (You know, children dig me rather.)
The following is not as informal. (Probably the sense of appeal was modeled on it.)
영업전술이 그 지역에 먹히다. (The sales strategy works in that district.)
As for a "personified" subject, we might have:
- 갈리아가 로마에게 먹히다. (Gaul is eaten by Rome.)
- 작은 회사가 대기업에게 먹히다. (A small company is eaten by a conglomerate.)
Note: I don't know that 갈리아 and 회사 are personifications. I am only noting a degree of agency in them; e.g. we may think of Gaul or a small company as not wishing to be "eaten."
Item 2 does not apply to these:
- 벽이 차에게 밀리다.
- 물건이 사람에게 발견되다.
- 모습이 그에게 보이다.
- 소리가 그에게 들리다.
As for the last two, there is little agency in 모습이 or 소리가 or 그에게. The sentence means either that the thing appears to the person or that the person is in a position to perceive. Moreover you cannot say:
소리가 그에 의하여 들리다. So they perhaps belong to an altogether different class than all the others.
This shoddy attempt at generalizing makes me think that there may be no alternative to picking up one word at a time as taking 에게.
I know this is annoying, and you'd like a rule that always works.
But you often do not have mechanical rules. Consider to forbid someone to do vs. to prevent someone from doing. A student of English will just have to accept this. Or remember to do vs. remember doing; what rule would correlated these to their respective senses? Or the way French verbs take an infinitive as mediated by à or de or as unmediated; you cannot deduce the result from your knowledge of English.
By the way, just about no one will say:
걔 남자친구에게 차였어.
although people would understand it.
You would say:
걔 남자친구한테 차였어.
One final point worth making explicit is that verb pairs in -다 and -이/히다 are thought to be separate words, listed separately in dictionaries (as opposed to conjugations of the same verb). E.g. from your sentences, 잡다 and 잡히다, 차다 and 차이다, etc. 들키다 does not have an active counterpart.
Per OP's comments, a few words on "agency." What I mean by that is whether some attitude or volition is implied for the thing said to have agency (or not).
The obvious contrast is between:
- 어린 양이 늑대에게 먹히다.
양고기가 늑대에게 먹히다.
Here, the availability of attitude or volition to the lamb, but not to mutton, is what makes the one sentence good and the other bad.
For what may be more tricky:
갈리아가 로마에게 먹히다.
Gaul is being compared to an agent, such as a person or tribe, that may have an attitude or volition about being ruled from Rome.
흑해의 자리에 있던 호수가 지중해에게 먹힌 날. (Literally, the day that the lake which used to be at the place of the Black Sea was eaten by the Mediterranean)
Now you're being downright poetic as if the lake did not like all that salt water rushing in.
The following is altogether different in that 에게 is "to" rather than "by."
소리가 그에게 들리다.
In terms of agency, nothing in this sentence has it. If you ask 내 말 들려? it means whether your voice (lit. speech) is audible to whomever you are addressing. If the person is properly situated, he has no choice about hearing or not hearing.