Is A Draw Enough For White in the Jerome Gambit?

Given that the Jerome Gambit (1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+) is a "refuted" opening, should White grab the chance for a draw if he sees one? Or does this go against the whole idea of playing a wild, destabilizing and gambling opening?

Is there a point where the Gambiteer should realize that things have gotten difficult enough that he should save a half point, rather than surrender the whole thing?

The following game is an interesting example of these issues.

deriver69 - golddog2
Giuoco Piano Jerome Gambit Tournament
RedHotPawn.com 2016.

1e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+

4...Kxf7 5.O-O Nd4

An odd move, reminiscent of the Blackburne Shilling Gambit; but perhaps not too odd: there are 10 previous examples in The Database, and White won 6, lost 4.

6.Nxe5+ Ke7

Better was 6...Kf8 7.c3 Nc6 8.Nxc6 bxc6 9.d4 Bb6 which is about equal according to Stockfish 7.

7.d3

Instead, after 7.c3 Ne6 8.d4 Bb6 Stockfish 7 says White has the advantage.

7...d6 8.Bg5+ Nf6 9.Ng4 Bxg4 10.Qxg4 Nxc2

Here we have a typical messy Jerome Gambit position. Stockfish suggests that Black should take the time to get his King out of the pin, with 11...Kf8.

11.Nc3 Nxa1 12.Rxa1

White overlooks (or ignores?) the fact that his sacrifice allows him now to draw with 12.Bxf6+ Kxf6 13.Nd5+ Kf7 14.Qf5+ Kg8 15.Qe6+ Kf8 16.Qf5+ etc. But - was he even looking for a draw?

12...c6 13.e5 dxe5 14.Re1 Bd4 15.Ne4

White continues to pressure the Knight at f6.

Black would do well to further consolidate with something like 15...Qd5. Instead, he decides to be a bit more aggressive - and this gives White an opportunity that he does not miss a second time.

15...Qb6 16.Bxf6+ gxf6 17.Qg7+ Ke6 18.Qxf6+ Kd7 19.Qg7+ Ke6 20.Qf6+ Kd7 21.Qg7+ Ke6 22.Qf6+ Kd7 23.Qg7+ Ke6 24.Qf6+ Kd7 25.Qg7+ drawn by repetition