The phrase "to come to power" is semantically analogous to "to be voted into office". Coming to power is not something someone does in the sense of performing an action. Rather, it is something that happens with respect to them.
If we say:
His wife caused him to be voted into office.
we would mean that his wife had influenced the voting in some manner. She did not act upon him directly but indirectly; she acted upon those who voted. It is similar to "She caused the house to be painted". The house isn't doing anything. The house is passive. She had someone paint the house. She influenced the election. She did not make him do something.
If we say:
His wife led him to be voted into office.
the meaning is not very clear at all. We can lead a lamb to be slaughtered, but can we really lead a person to be voted into office? She might have led his campaign. She might have led him to victory. But did she lead him to be voted for? This is semantically a marginal case. It is not really idiomatic. We normally "lead someone to do something", and being voted for is not something a candidate can do.
The same is true with prompt. We prompt someone to do something. For example, we can prompt someone to volunteer. But we cannot prompt someone to be tattooed. We can prompt them to get a tattoo, or to allow themselves to be tattooed. There must be a volitional element in the complement, something the person who is prompted is capable of doing.
Consider:
His wife made him (to) be voted into office.
That too does not make good sense. The pattern is used in contemporary English like this: "to make someone {do something}". The complement of make ('him') is the one who performs the action in the infinitive phrase. But being voted for is not something he can himself do. In centuries past, however, made s.o./s.t. to be + pp. was used like "caused ... to be". But we don't use the verb in that way nowadays.
