In theory, optimal policies could be tabularly implemented. In this case, it is impossible for them to further improve their “planning.”
That sounds wrong. Planning as defined in this post is sufficiently broad that acting like a planner makes you a planner. So if you unwrap a structural planner into a tabular policy, the latter would improve its planning (for example by taking actions that instrumentally help it accomplish the goal we can best ascribe it using the intentional stance).
Another way of framing the point IMO is that the OPs define planning in terms of computation instead of algorithm, and so planning better means facilitating or making the following part of the computation more efficient.
That sounds wrong. Planning as defined in this post is sufficiently broad that acting like a planner makes you a planner. So if you unwrap a structural planner into a tabular policy, the latter would improve its planning (for example by taking actions that instrumentally help it accomplish the goal we can best ascribe it using the intentional stance).
Another way of framing the point IMO is that the OPs define planning in terms of computation instead of algorithm, and so planning better means facilitating or making the following part of the computation more efficient.