うまくいく

Do native speakers generally think of うまくいく as a verb phrase or a full-fledged verb? I’ve noticed that うまくいく is paired with adverbially inflected i-adjectives instead of the continuative form–i.e., 多くうまくいく not 多くてうまくいく. If うまく was parsed as the adverb form of うまい, I would expect the latter, not the former.

Here’s a contextual example:

あなたのことだから間違いなくうまくいくと思うよ

A quick Google search shows 1400+ examples of ~くうまくいく and only three of ~くてうまくいく.