Trading Down Against the Jerome Gambit

If someone makes an exchange and comes out ahead, that is considered "trading up". Do the opposite, and that is considered "trading down". Why would someone want to do the latter against the Jerome Gambit?

Wall,B - Guest2115687
PlayChess.com, 2014

1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+ 

4...Kxf7 5.Nxe5+ Nxe5 6.Qh5+ g6 7.Qxe5 Bxf2+

As I've written elsewhere,

Just about everyone who plays the Jerome Gambit faces this "bail out" variation at one point or another. Black manages to exchange Queens at the cost of returning a piece, and with the prospect of playing on, a pawn down. White can no longer play "his" attack; but he also no longer has an "objectively" lost game.

8.Kxf2 Qh4+

Last year Bill's brother, Steve, faced the more direct 8...Qf6+ 9.Qxf6+ Nxf6 in Wall,S - Guest658246, PlayChess.com 2013 (1-0, 34).

9.g3

A couple of years ago Wall,B - Guest1443273, PlayChess.com 2012 (1-0, 36) continued 9.Kf1 Qf6+ 10.Qxf6+ Nxf6 etc.

9...Qf6+ 10.Qxf6+ Nxf6 

11.Nc3 Re8 

Instead, Bill suggests 11...d5.

12.d3

Threatening 13.Nb5 and Nc7 

12...d6 13.Bg5 Ng4+

He would do better to concentrate on development with 13...Be6.

14.Kg2 h6

Or 14...c6 15.Rhf1+.

15.Raf1+ Kg7 16.Bd2 Bd7 17.Nd5

Threatening 18.Nxc7, forking the Rooks. 

17...Rac8 18.Bc3+ Kg8 

Now White offers one final trade, which turns out to be deadly.

19.h3 c6 20.hxg4 cxd5 21.Rxh6 

Threatening 22.Rh8 mate. Black resigned.