Black has better than 4...Ne8 : 4...a5 and if 5.Bxa5 Ra6 gets out of the pin. Of course, White has one more exchange and two connected passers, so he should win. But maybe better is even (4...a5) 5.Be1 ! after which 5...a4 6.Rf2 axb3 7.axb3 is only a very tiny improvement (Black is still dead lost).

I don't see the advantage of 4... a5:

4…a5 5. Be1 a4 [6. bxa4 Bxa4 7. Rxc6 Bxc6] 6. Rf2 axb3 7. axb3 [Bxb3?? 8. Rxc6 1-0]

I don't see why you give Be1 a "!", but sure. Rook connectivity isn't too key here, it's not that noteworthy of a move. Black trades pawn for pawn, giving white a passed B-pawn. Seems bad.

Actually, the b passed pawn is not a key factor here. Black is trying everything he can to avoid losing a piece. If he can get a try to swindle White even at the cost of a pawn, he must take it.

Ok, 4...a5 is not "better", it's only more tricky. At least a better try than 4...Ne8.