One thing people seem to forget or not even know is that, at that time, the POTUS typically responded to requests from State Governors in regards to FEMA resources. Also guys like New Orleans Mayor Ray Nagin simply waited too long to direct orders, and didn't use buses due to insurance issues (among others). Amtrak offered to shuttle civilians out as they were rerouting equipment from the area and local officials declined. A lot of this will still fall on the POTUS, especially when the "media" can make a big issue over the VC-25A flyover when Bush could have gone in at ground level.
I am not sure how much was "media" then and how much was procedural garbage or poor POTUS response but I imagine it was a decent mix of all three. In 06 I didn't really watch any of the "news" and was barely starting my business so I have no verifiable archives to reference. Actually just before Katrina hit I was out at Coronado and Bush was near, or at the area, I can't really recall the exact dates.
This report covers all changes to the post-Katrina act, Title VI of P.L. 109-295 (H.R. 5441) broken down by each revision and isolated to Congressional Acts:
https://www.everycrsreport.com/reports/RL33729.html#_Toc229817903 These changes altered a lot of how FEMA works and the authorizations the POTUS has. The Hurricane Sandy Recovery Improvement Act, P.L. 113-2 made even more changes including FEMA was directed to update its criteria for considering whether to make a recommendation to the President for Individual Assistance declarations.
There are still hearings on this due to the politics involved in how a POTUS may or may not declare a disaster.
List of Authorities and References:
https://www.fema.gov/pdf/emergency/nrf/nrf-authorities.pdf