grammaticality - What do you think about "Sorry." as a complete sentence? or What are your thoughts on subject omission?

I've been poring over materials on Japanese (日本語) and found it common of them to contrast the language with English in saying that pronominal subjects can be —and typically are, as with 私は (Watashi ha, basically I)— dropped in Japanese, whereas with English subjects cannot be elided. A cursory evaluation, as an English speaker, of the truth of that proposition didn't raise any objections. However, upon thinking on it further, I was able to imagine many examples of common English usage where that doesn't seem true, wherein the subject is omitted.

P₁: We'd've done well to've left earlier. We'll be in this traffic for hours now.
P₂: Sorry.
vs
P₁: ”
P₂: I'm sorry.

I'm especially interested in this example since it's not a response to a question. With examples that are responses to questions, I see how a point could be made that the relationship between the question and its subject-less response is somehow anaphoric —the subject expressed by the question is projected forward the the answer so that the answer implies it. Whereas, statements that prompt "Sorry." sometimes lack, I think, an antecedent to imply an "I" in the response.

or

P₁: How've you been?.
P₂: Well.
vs
P₁: ”
P₂: I've been well.

or

P₁: Heading out; need anything?
vs
P₁: I am heading out; do you need anything?

Please share your take on any dimension of this. Thanks.