You could have a passive participle, but that is only possible for a transitive verb (so you can have "John was arrested/eaten"). "Died" is only a past tense verb, and that would mean that tense is instantiated twice in the same clause (you have tense only once).
This
is a past participle (or past tense), as in "John died" (past tense) or "John has died" (past participle). But the first part of your explanation is correct there: the participle must be from a transitive verb.
Passives promote a non-subject to subject, and remove the subject as an argument of the verb (it may appear as an extra bit of information then, introduced with "by").
So the verb "die" does not have enough arguments (just one: no object like "John died his life"), meaning that there is nothing to promote to subject.
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Now, you might wonder, what about not having a subject all? Well, that's generally not allowed, and at least in English you would need to fill in something like "It was died". Even in a language like Spanish where overt subjects are not required, there is still generally an understood subject (as indicated by subject agreement still found on the verb!), so that also wouldn't make sense: there must be a conceptual subject, even if it is not pronounced.
Interestingly enough, there actually are some languages that do allow passivization of intransitive verbs. In German, the verb "werden" (lit. 'become') works like English "be" in passives. And you can literally say "It was danced":
Es wurde getantzt. -- meaning something like "There was some dancing going on; people danced." I'm not sure whether "Es wurde gestorben" would make sense to a German speaker ('there was dying going on'?), but it may be a grammatical possibility. But again, English does not allow for this.
