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?
It's good but "橋本聖子さんに会長になるように頼む" is more natural.
Is 橋本聖子さんに会長を頼む just a shorter way of saying this?
Yes. 頼む can simply take a noun, e.g. 寿司を頼む (order sushi / ask them to make sushi) and あとを頼む (leave them to do the rest). Usually we don't distinguish 'presidency' ...
Generally, Japanese song lyrics tend to be loose in grammatical integrity, and much more for those whose lyricist is Hata Aki (I mean!).
While sung, the parsing seems like 夢を持ってきた言葉じゃ / 足りないユメのアリカ, with a pause after じゃ; I understand that could just be for musical reasons
Regarding this, all lyrics I find make a line break between 持ってきた and 言葉, so I think ...