Inflight restarts are hard. Starship has several hurdles. The Raptor is a pump-fed engine, and turbopumps need a sufficiently high, sufficiently stable head pressure. Starship has highly exciting flight dynamics just when you need liquids not to slosh in various tanks and pipes.
Previously, inflight restarts were obviated by having a hypergolic upper stage and bladder-pressurized propellant tanks. No pumps to cavitate, no igniters to fail. But hypergolics bring their own engineering problems with them. And the available chemistry limits power and efficiency. So restartable pump-fed or pressure-fed engines, with chemical or electrical ignition, has to be part of the vocabulary. As with Apollo, pump-fed engines can be restarted by an ullage burn to "settle" the tanks, and a spark plug that ignites the preburner using propellants fed to it only by inertia from the ullage. This works best when the vehicle isn't otherwise maneuvering. And sometimes it involves several stages of ignition (i.e., using small igniter torches) based on how much propellant flow is possible from the previous step. And multi-stage ignition sequences generally have to be very precisely timed.
Now complicate the fluid slosh by having your vehicle do Olympic diving maneuvers. Head pressure fluctuations, gas voids, pump cavitation -- so much can go wrong. It looks to me like the turbopumps just weren't getting a bite on the propellants, for whatever ultimate reason. When they say they're still working out the mechanics of Raptor restarts, I don't doubt it for a second. What they're trying to do is very hard. If I have some time later, I'll look into the Raptor restart sequence and the Starship propellant feed system in more detail.
The two debris sheds look like the insulation blankets around the upper landing struts. The debris looks like it's coming from the perimeter of the skirt, not the center. And it's not the struts or footpads themselves -- at least the first one -- because all six footpads are still in place after the first shed. Plus, the ease with which the slipstream slows them down suggests lightweight film, not heavy structure. Losing insulation in the last five seconds of the flight would be inconsequential. It may be planned for in SpaceX's flight profile. I would be interested to see the slipstream turbulence models for the interior of the skirt.