This sounds like a good idea for a plugin, but there's nothing out there that does it.
Basically, you want your product owner to keep on top of things. One way to do this is to flag "unhealthy" issues in a JQL filter.
If you make a custom field for your acceptance criteria, it would be easy to highlight issues for which this field is empty
"Acceptance Criteria" is EMPTY
How far in advance are stories sized is trickier. You would need to ask "Up to which Rank value do backlog items have Story Points", but the ranks are not stored as 1,2,3..., they're lexorank values (1|asdfds, 1|dhaseld....) that are handy for ordering, but not so much for filtering.
So, you'd need something else to identify the items that are due to be started soon. That depends on how the PO works. If the PO uses due dates or any other date field, you could do something like this:
("Acceptance Criteria" is EMPTY OR "Story Points" is EMPTY) AND "Due Date" <= 14d
That query means "issues due 14 days or less from now that have no story points yet or that don't have acceptance Criteria".
A way to obtain a metric would then be a count of the number of issues that require the PO's attention
This is obviously a complex topic, but any solution you might reach will involve the PO working in a specific, trackable way, using statuses, votes, filling a future "Ready to plan" sprint with "healthy" issues ahead of time etc.
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