If there is an polynomial time approximation to an NP-complete problem, is P approximately NP?
No, it doesn't prove that P = NP.

This has nothing to do with approximations; it has to do with average-case hardness vs worst-case hardness.  The two results are showing that solving the problem for a randomly chosen hypergraph is usually easy; but there exist hypergraphs where the problem is hard.  Presumably, choosing a hypergraph at random is very unlikely to give you one that's hard, but they do exist.

NP-completeness is about worst-case hardness: even if a problem is easy on average, it can still be NP-complete.

I have no idea what "P is approximately NP" would mean in any precise sense.