What exactly bothers you in this line? According to the database there are different ways for Black to play the position. Maybe White is a tiny bit better because the Nc6 is not ideally placed but I don't think that Black has problems in this symmetrical structure.
There are many ways to play with black, but in all of them, I think white gets a slight advantage due to his better placed pieces. Also, statistically it scores very well for white, it feels like one of those positions where white is playing for 2 results. I was thinking maybe black has a more clear cut equality in the other 5th moves.
Edit: I meant 6th move, same goes for the original post
If memory serves me right, back in the day when I played this (I only play the White side now and I play 3.d4), Black is supposed to play 6...Be7 and delay the development of the Knight. If White plays 7.Be3, the Knight should be developed on c6. If White plays 7.Bf4, then the Knight should be developed to d7 with intentions to play ...Nc5.
If memory serves me right, back in the day when I played this (I only play the White side now and I play 3.d4), Black is supposed to play 6...Be7 and delay the development of the Knight. If White plays 7.Be3, the Knight should be developed on c6. If White plays 7.Bf4, then the Knight should be developed to d7 with intentions to play ...Nc5.
I don't see Be3 or Bf4 being played in the 5.d4 line that early on. Are you maybe confusing this with the 5.Nc3 Nxc3 6.dxc3 line?
In one of the mainlines of the petrov (1.e4 e5 2.Nf3 Nf6 3.Nxe5 d6 4. Nf3 Nxe4 5.d4) I see that there are mainly 3 move being played, 5...Nc6, 5...Bd6, 5...Bf5. Currently, it looks like 5...Bd6 is the most popular with Caruana playing it. Even so, I'm still leaning towards 5...Nc6, but I don't like this line 1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. d4 d5 6. Bd3 Nc6 7. O-O Be7 8. Nbd2. What are your thoughts?