I agree on the first one: as a President candidate, he might have made himself a service by putting his full name on the thing; I suppose it would make him more serious.

Childish font? Certainly not. Serif fonts like this are used in the design and literary works, most of which I'm sure you'll agree are quite adult things. Granted, it doesn't fit there - though I must admit, I don't know Jeb Bush as a person, maybe he is a guy like that - but it's not a reason to give it a label it doesn't deserve.

Same goes for the exclamation mark. It doesn't fit the seriousness - or, supposed seriousness, for all I know (I have no idea about the US elections) - and the most of such marks we can see today are in the old entertainment media, like Batman comics or movie posters. No reason to call it what it's not.

2016 is placed very well, in my unprofessional opinion. Given a sans-serif font and the right sizing, it might have a good looking at the current logo. Though, given a sans-serif font, I'd put it at a line lower than the baseline of the name, at the right corner (right under "b" in the current logo). Then again, I have no idea what kind of image does Jeb Bush put forth, so I can't judge properly - only build on abstracts.

So, "un-presidential"? Probably. "Ugly", though? Obama's "HOPE" was far less good-looking than the slicker, simpler design of the Jeb Bush logo. It's not perfect, but what it's worth, it's a good start.