Surviving re-entry will be one of the initial critical aspects of the Starship development program.
According to a statement by Elon Musk during his recent interview with Tim Dodd (Everyday Astronaut), they are not worried about re-entry yet. Elon Musk said they don't really know whether they will even get to space, let alone orbit. It is entirely possible that the vehicle breaks up on ascent. In fact, he said, he would consider anything that does not destroy the pad a success.
If there are locations on the vessel where the heat-shield fails, leading to its destruction how would SpaceX be able to receive any data during this phase of flight to diagnose the failure?
It is not necessary to receive data during this phase of flight to diagnose the failure. There's no deadline, they can just as well receive the data a bit later, e.g. from a flight recorder.
Also, a lot can be reconstructed by inspecting the wreckage, if they really wanted to go to the trouble of doing that.
I understand that radio communications are not possible when the ship is surrounded by plasma.
It is not necessarily surrounded by plasma. The shockwave will be in front of the vehicle (where "in front" means "roughly along the velocity vector"), and that's where the highest concentration of plasma will be.
So, there is a good chance that they will still be able to send data "up", just not "down". As it so happens, there are about 1200 SpaceX-owned communication satellites in a low earth orbit, plus they have access to NASA assets.
That's what the Space Shuttle Orbiter did in the later years, for example. Instead of communicating through the plasma to the ground, it threaded the needle through a "hole" in the plasma created by the airflow around its body upwards to the TDRSS network.