Not Commas But Not Not Commas, Either

I'm pondering the commas in this sentence, which was is based on a passage from a CNNmoney.com article:

"Then someone sees, not words on a page, but a vibrant person sitting across the table from them."

From time to time I encounter these structures while copy editing -- the "not blank but blank" structures. I'm never confident in how to handle them commawise.

Should "not words on the page" be set off with commas, as it is here? Clearly, the phrase is an insertion -- an interruption. But, combined with the "but" before "a vibrant," the commas sever the transitive verb (sees) from its likely intended object: (a vibrant person).

Obviously, there's nothing wrong with inserting parenthetical information between a verb and its object. But when the result is to resume the non-parenthetical with a "but" that comes before the object of the verb, it seems to create a logic problem for the whole structure.

My inclination is to leave the commas out: "Then someone sees not words on a page but a vibrant person sitting across the table from them."

I should note: The original sentence was actually a little more complicated: "If someone sees, not words on a page, but a vibrant, energetic person sitting across the table from them, age becomes much less of an issue."

Those other, obviously correct commas seem to somehow make commas around the "not blank but blank" phrase more beneficial.

If only this issue were as easy as whether to eat turducken.