Is King's gambit really an inaccuracy

After 2..e x f4 3.d4 is not really safe..Nf3 is better. It takes away the h4 square from the black queen.

3 d4?! Qh4+ 4 Ke2! is playable.
3 Bc4 Qh4+? 4 Kf1 is good for white.

The white King is uncomfortable, but the black queen even more so. White gains time with developing moves that attack the queen, starting with 5 Nf3.