You have truncated my original examination of the phrase, maverick. I did not stop at: “I ‘am in possession of’ got to go.” That was merely the jumping-off point. I continued the expansion of the phrase to its possible conclusions, as follows:

I ‘am in possession of’ ‘came into possession of’ to go. or
I ‘must’ ‘came into possession of’ to go.

Nevertheless, your truncation aside, let’s examine the phrase at the point at which you chose to stop:

I ‘am in possession of’ got to go.

Using the definition you cited, the next possible expansions would be:

I ‘must’ ‘have an obligation’ to go. or
I ‘am in possession of’ ‘have an obligation’ to go.

The expansion: “I ‘am in possession of’ ‘have an obligation’ to go” is more faithful to the model I originally established of replacing words with their definitions.

Your conclusion of: “I am possessed by an obligation + to go” takes the expansion of : “I ‘am in possession of’ ‘have an obligation’ to go” and then tries to make it more understandable by changing the definitions of the words ‘have’ and ‘got’. You have changed the definition of ‘have’ from: ‘am in possession of’ to: ‘am possessed’. You have also changed the definition of ‘got’ from: ‘to have an obligation’ to: ‘an obligation’. Changing the definitions of the words was not part of my original model because it is pointless to do so and unnecessarily clouds the issue.

Looking at the two expansions above, the phrase either conveys a different meaning altogether from what is understood by ‘I’ve got to go’ or it requires further expansion.

“I ‘must’ ‘have an obligation’ to go.”

Here the meaning is different. It makes one wonder, “Why must you have an obligation? Is someone forcing you?”

“I ‘am in possession of’ ‘have an obligation’ to go” expands further into:

I ‘am in possession of’ ‘”am in possession of” an obligation’ to go.

Here the meaning is redundant. There is no need to repeat ‘am in possession of’.

You asked: “Is that really so awkward?”

Let’s compare your redefined expansion with the most clear and concise way to convey the meaning of ‘I’ve got to go’:

I am possessed by an obligation + to go

I must go.

I don’t know about ‘so awkward’, but yes, without a doubt, it is awkward – unnecessarily awkward.

You mentioned ‘meaning’. Yes, speakers and listeners can invest meaning in whatever they agree upon. But why make communication unnecessarily awkward? As I originally wrote, “Wouldn’t it be clearer and more concise to say, ‘I must go’ …?” The same meaning would be communicated without all the ambiguity.

Generally speaking, I think tswum has given the best explanation for something so ambiguous being used so widely and popularly: idiom.