I think you misunderstood both the position and Naroditsky's comments.
Stockfish says starting position is losing.
The point of the study, as often, is that the correct evaluation goes against your first intuition. Here Stockfish is confused by the huge material advantage Black can gain and typically fails to assess correctly that White can build a fortress and save the draw.
The credit for this gem goes of course to the study's creative author, Vitaly Chekhover, not to Naroditsky.
That's what Daniel also wrote in that book.
Daniel Naroditsky did not write that White is losing. He wrote "At first sight, White is completely lost" which hints that there is more to be found than what's on the surface and stresses out how miraculous the saving idea will be.
If black doesn't play Re4 then white is likely to out of moves sooner (will be in zugzwang.)
That's not correct. White has the waiting moves Kf1-f2-f1 when the black rook is on the e-file and Kg1-g2-g1 if it switches to the h-file. Thus White will never be in zugzwang and can patiently hold in his fortress. Whatever the evaluation of your engine, this position is a draw.
After Re4 it is winning for white.
Even if it's true it won't change Naroditsky's point: with or without ...Re4, Black cannot win.
Didn't Daniel calculate other variations while analyzing this position (I am not asking if he had really did. But I am asking why didn't he show other variations as he did for other examples)?
The position after 9.a4 is a positional draw where there is no variation to be checked at all: ...a6 or ...a5 or king moves are pointless, the white King can deny any entry square from the black Rook, so Naroditsky analysed 9...Re4 because targetting d4 is basically the only attempt to disturb the balance.
The reader is expected to check by himself that "Black has no way to break through", you can try to shuffle the rook a bit to convince yourself, but no extra variation can help you reaching that conclusion. He could have given a line like 9...Rh6 10.Kg2 Kf6 11.Kg1 Ke7 11.Kg2 a6 12.Kg1 Re6 13.Kf1 a5 14.Kf2 Kd7 15.Kf1 Re8 16.Kf2 and so on ad nauseam, but that would not have been much more informative.