grammar - Can someone please explain the syntactic rules at work here?

I'll use an example statement that's currently being used in a radio commercial for American Family Insurance (paraphrased.)

They all told me that I couldn't build my dream home by myself; but, I didn't.

Can someone please lex this to identify why but, I didn't is referring to the fact that the dream home was built and it was not build by myself; whereas and, I didn't would introduce ambiguity as to whether the dream home was built or not; or that it was built but not by myself.

To state my question another way: if I were to attempt to automate a lexical analysis of this statement -exactly as it has been provided- to conclude that the house was built with the help of others, what rules would I need to consider?

Apart from bare assertions that the sentence parses a different way, can anyone see any alternative interpretations?

[Edit]

I see the but as an adverbial conjunction and that's why my parsing is conflicting with that of others who are seeing it as a coordinating conjunction.

They all told me that I couldn't build my dream home by myself, only I didn't.

With this understanding, it is clear to see that this is the structure they are using. The question now is...is that the proper structure of an adverbial but? It feels natural to me, but by the responses below, others seem to disagree.