The obituary for Hilbert's Program is premature to say the least. It does not rule out significant partial realizations. This is a remarkable vindication of Hilbert. The need to defend the integrity of mathematics has not abated. The assault rages as never before.
Platonic intuition is unsuitable as a weapon with which to defend the validity of mathematics.
Only the first few levels of the cumulative hierarchy bear any resemblance to external reality. The rest are a huge extrapolation based on a crude model of abstract thought processes. Unfortunately such tests seem hard to carry out. Finitistic reductionism is not the only plausible method by which to validate infinitistic mathematics. One might try to show that a substantial part of infinitistic mathematics is directly interpretable in the real world.