I agree; it's not safe to assume the application "didn't go through". I would wait to see a notice of abandonment. Even then, the application can be revived under certain conditions, for example, when the delay in reply by applicant or patent owner was unavoidable or unintentional.
You should not assume that the application has been abandoned. The language you quoted refers only to the limited time within which the applicant can file a statement summarizing the substance of the interview. Absent such a statement from the applicant, much of the substance of the interview will not become a part of the file history. This one-month deadline is distinct from any other deadline imposed for responding to substantive rejections, which generally have a statutory reply period of six months.
I agree. It is not safe to assume that the applicant has abandoned the application. You will need to wait for the PTO website to be updated to confirm any abandonment. You should also be aware that a related application could currently be on file or filed before the application you are watching is formally abandoned.
Disclaimer: This answer does not establish an attorney-client relationship and does not constitute legal advice. It is for general information purposes only.
An applicant is not required to respond to the Interview Summary. One would do so if you disagree with the Examiner's summary of the discussion.
That interview was almost certainly preceded by an Office Action. Failure to respond appropriately to the Office Action would lead to abandonment of the application. Even still, the application could be later revived for a fee payment if the applicant abandoned the application either unintentionally or unavoidably.
It is definitely not safe to assume that the patent application didn't go through.
Do not assume that the application did not go through. It is often months before the next action takes place, and even then (as you mentioned) the USPTO website often takes time to update. Notably, failure to submit an interview summary does not constitute abandonement of the application. Further, even if the claims continue to be rejected (and assuming this is on final rejection), the applicant could file a request for continued examination or follow a number of other administrative routes.