Checked the current Agent mode Responses API docs on this (docs.databricks.com/aws/en/genie-agents/api) and your read looks right, not a gap in your search.
The response stream only documents four output item types: reasoning, function_call, function_call_output, and message. None of them carry text_instructions, example_question_sqls, or relevant_tables the way progress_report.context_info does in standard mode. So as things stand there's no documented field or event type that reports per-turn context in Agent mode.
One partial workaround: function_call items expose the literal execute_sql arguments, i.e. the SQL that actually ran. You can diff that against the space's configured example_question_sqls and table list (from serialized_space) to infer which example/table likely influenced a query. That's inference after the fact though, not a record of which instruction fired, so it won't catch your "missed instruction silently propagates across 5+ queries" failure mode.
Given Agent mode APIs are still Beta, this seems like exactly the kind of gap the Genie team would want flagged directly, e.g. via the in-product feedback widget or a support ticket linking this thread, since whether it's roadmapped isn't something the docs will answer. Tagging in case anyone from the Genie team is watching this board and can confirm either way.