It's just not a good practical choice because you would have to move like an engine with black to not be blown away. See Shirov vs Sulskis 2014 as an example of what can happen. The game follows your Stockfish continuation until Nb4 but Shirov goes a3 immediately instead of Bb3, sacrificing the rook after Nc2+. Perhaps Stockfish can defend this but Sulskis could not over the board.
Is 5...Nxd5 bad in the fried liver attack?

Yes because there is a far superior move in the two knights defense, not allowing the fried liver attack.
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 d5! 5.exd5 Na5! and the position is already equal! Why allow the fried liver attack when you can prevent it immediately with the main line of defense against the Ng5 variation of the Two Knights Defense?
Many mistake 4.Ng5 as the "Fried Liver", it is not. The Fried Liver is specifically 6.Nxf7 after 5...Nxd5?! (That is being generous calling it dubious, BAD is more accurate!)
As far as I know it has been known for centuries that 6.d4 rather than 6. S*f7 is even stronger . For example Tarrasch in his manual "tha game of chess " from 1930 states : '6 d4! clearer than the very interesting knight sacrifice on f7... '.
And Morphy often played a differnt move order to reach the d4 position. I think it was first Giambattista Lolli in the first half of the 18th century who introduced 6.d4.
To cut a long story short: It has been known for a long time that 6. d4 is probably stronger, but from a practical point 6. N*f7 might cause more problems for your opponent if he isn't prepared and you are.
I have just read the above-mentioned article on chess.com, so it seems that deep computer analysis has reversed the evaluation and N*f7 might be even stronger.

The lolli attack (d4) is stronger but it requires white to know the theory, in the fried liver black has to know very well the theory to survive

Yes because there is a far superior move in the two knights defense, not allowing the fried liver attack.
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 d5! 5.exd5 Na5! and the position is already equal! Why allow the fried liver attack when you can prevent it immediately with the main line of defense against the Ng5 variation of the Two Knights Defense?
Many mistake 4.Ng5 as the "Fried Liver", it is not. The Fried Liver is specifically 6.Nxf7 after 5...Nxd5?! (That is being generous calling it dubious, BAD is more accurate!)
That's called poleiro defense and it's the main and best line against the knight attack

It's not as bad as it's made out to be. However it is bad, bad enough to be worse than the Ulvestad and the related lines (b5 or Nd4 and so forth), let alone a lot worse than Na5, especially in practical chess, and that means you should avoid it.
Unless you're this fellow lol!
https://www.youtube.com/watch?v=BQwEyr5qwws
It is my understanding that the "correct" 5th move for black in the fried liver attack is Na5. Doing Nxd5 is considered a mistake.
It's considered a mistake because of the following 6.Nxf7 sacrifice. And, in fact, if you look in chess opening databases, this line has lead overwhelmingly to white victory (I think over 80% of the time).
However, when I analyze 6.Nxf7 with Stockfish9... it doesn't actually consider it a winning move. No matter how long I let it evaluate, the evaluation remains about dead equal. In fact, if I analyze the previous position, Stockfish quite quickly suggests a completely different move, namely d4 (which itself also evaluates to a quite even position).
One possible continuation that Stockfish suggests is this:
Stockfish evaluates this position as about +0.5.
I suppose that in order to pull off 5...Nxd5 successfully, you have to be really accurate if your opponent goes for the Nxf7 sac. You have to know how to deal with your king in the open.