As for not being able to give an accurate estimate, that is not for the on-the-radio approach controller to calculate, given their view of the airspace. The video posted by parent shows how long the inbound flows were (at least on the east side); approach wouldn’t have seen that.
The coordination necessary to get an accurate estimate should’ve involved managers, which might be affected by the current shortage (see https://www.npr.org/2023/11/09/1211838624/air-traffic-contro...).
