Thank you very much, Maurits. It is encouraging that you agree that 6...Nf6 7.f5! offers sufficient chances. After 7...Be7 8.h3 I had considered 8...d5 only superficially. Overall I like White's chances after 7.f5, at least from a practical point of view: the advance of pawns gives White a lot of space, and Black has to generate counterplay, cannot just wait.
So maybe
1.e4 e5 2.Nc3 Nc6 3.d4 Nxd4 4.f4 d6 5.Be3 Nc6 6.Nf3 exf4(!) 7.Bxf4 is indeed the critical line.
(a) If
7...Nf6, 8.Qe2! (Maurits Wind) is probably the most accurate reply. Putting a bishop to c4 makes no sense, as the attacks on this piece (on c4 or b3) will only accelerate Black's play on the queenside. Wind's analysis 8... Be7 9.0-0-0 0-0 10.h3 Be6 11.Kb1 Re8 12.g4 a6 13.Bg2 looks fine. White has a space advantage and free piece play for the pawn.
(b)
7... Nge7 8.Qd2 (or 8.Qe2 Ng6 9.Bg3 Be7 10.0-0-0 Bf6!)
8...Ng6 9.Be3 (my first idea was 9.Bg3 Be7 10.0-0-0, but Black plays 10...Bf6! followed by Be5 and wins a tempo, because White doesn't really want to exchange on e5)
9...Be7 10.0-0-0 0-0 11.Nd5, with active play for the pawn, e.g. 11...Bf6?! 12.Be2 Re8 13.Rhf1! and pawn e4 is indirectly protected.
Whether 5.Be3 is really better than our former main line
1.e4 e5 2.Nc3 Nc6 3.d4 Nxd4 4.f4 d6 5.Nf3!? Bg4 6.Be2 Nxe2 7.Qxe2 c6 8.h3 Bxf3(!) Markovich 9.Qxf3 Nf6, is not easy to assess. For example, White could continue
10.Bd2 (to discourage Qa5)
10...Be7 11.0-0-0 Qc7 12.Kb1 0-0-0 13.Be3 Kb8 14.Rd2 Nd7 15.Rhd1 Qa5 (15...exf4 16.Bxf4 Nb6 17.b3 Nc8 18.Na4)
16.f5 Nb6 17.Rd3, and it is hard to see how Black can ever "free" his passive Be7.