Earlier quoted context omitted.
I think your example highlights one of the amusing things about Japanese. That is the complete omission of the subject. As you said, the 「私は」 might only get thrown on as an afterthought. You can have entire conversations without ever actually voicing the subject. It can come off like a British comedy sketch for somebody to come into the middle of an exchange and try to figure out the topic from the last half of the c…
> Edit: I should say that I never noticed a native Japanese person to be confused about the unspoken topic. They seem to be able to intuit it just fine in any context. I'm mostly just laughing at my own inability there. Sometimes they will get confused and clarify with a question: 私って? 誰のこと? 私とは、どちら様のことでしょうか?
Interestingly, it has the effect on me of giving it vocal inflection which works quite well in the context of the example.