Actually, I have been looking a lot at the Black side of the open games during the last weeks. I am not too impressed with the linked videos because he spends most of this time at the line 3... Nf6 4. d4 exd4? He does a good job of presenting the basic ideas, though.
The critical Black answer is 3... d5 which can lead to extremely tactical play with sacrifices in a lot of lines, so the sensible choice for Black (if he doesn't want to learn all this theory by heart) is probably 3... Nf6. After 4. d4 Nxe4 5. d5 Ne7 6. Nxe5 Ng6 we have reached the starting position of the line (which is only shown during the last few seconds of video 1).
Black's position is totally fine here (in spite of White's space advantage) as he is a bit better developed and White has some problems with developing the queen side.
I have used the Ponziani myself in some blitz games, and if Black knows what he is doing, the opening is as harmless as it sounds.
If someone is looking for an easy to play, sound opening after 1... e5, I would rather recommend the Bishop's game (or the Vienna move order) with f4.