The committee decided to ask Hashimoto to become president
I found this phrase a bit confusing. Both verbs in this sentence can take both に and を at the same time. I think に must go with 頼む in this case since XにYを決める means "decide on Y as X" and "decided on asking the president as Hashimoto" makes no sense.
That leaves me with how to understand 橋本聖子さんに会長を頼む. So XにYを頼む means "ask/request Y of X" and I get "request president of Hashimoto". That makes no sense. I think I'd have been happy with 橋本聖子さんに会長になることを頼む. Is this grammatical/natural? Is 橋本聖子さんに会長を頼む just a shorter way of saying this? Have I completely misparsed everything?
So assuming I have understood this correctly what kind of objects can 頼む take?
An after thought:
If I translate 会長 as "presidency" rather than "president" then all my problems go away. I wonder if this is the way to think about it.