Can an English sentence have a 'dative subject'?

Solution 1:

This is a fairly well-known phenomenon in English grammar;
the simplest explanation is that there are two syntactic processes interacting here.

• One process (rule, construction) is the Passive,
  which exchanges the subject and the object of a transitive sentence,
  without changing meaning -- but adding or subtracting a prepositional phrase:
  The janitor painted the fence. ~ The fence was painted (by the janitor)

• The other process (rule, construction) is the Dative Alternation,
  which exchanges the direct object and the indirect object of a bitransitive sentence,
  without changing meaning -- but adding or subtracting a prepositional phrase:
  The janitor sent us the bill. ~ The janitor sent the bill (to us).

These processes are optional: one can occur, or the other, or both,
as long as their conditions are met. Or neither. So there are variations:

1. He gave a gift to me.
   (no Dative occurs)
   == Passive ==>
   A gift was given to me by him.
   == Agent Deletion ==>
   A gift was given to me.

2. He gave a gift to me.
   == Dative ==>
   He gave me a gift.
   == Passive ==>
   I was given a gift by him.
   == Agent Deletion ==>
   I was given a gift.

Both are correct; they're just variations. The easiest way to look at it is that an indirect object can get promoted to direct object by Dative, and a direct object can get promoted to subject by Passive. If something's already been promoted once, it can still get promoted further.

The point (the communicational purpose) of the Passive is to downplay the agent subject and get some affected patient object up front for emphasis. Having both these variations allows any noun with a grammatical relation to the verb (Su, DO, IO) to get moved up front, where the action is. It gives the speaker more latitude, and obviously is adaptive, since it's universal in English (though those prepositions will vary somewhat, as usual).