Is black doomed?

Yes.

You still haven't told us what you hope to accomplish with 3. ... Bb4.

What do you hope to achieve with the move?

Statistics don't measure an openings strength.

A total absence of facts also doesn't measure an opening's strength.

You haven't offered either specific lines or specific ideas to support the 3. ... Bb4 move.

Not all gambits offer increased activity, although lots do.

Being a pawn down dimishes central control, as having less pawns makes it harder to control the centre, being the Evans gambit's major downside.

Well, you were talking as if giving away that Pawn was automatically advantageous to White.

It isn't. It wins time (by luring the Bishop to the vulnerable b4 square) and allows White to quickly build a center with c3 and d4.

If Black plays 3. ... Bb4, White gets those advantages WITHOUT the need to give up a Pawn.

Why is it poorly defended?

Because White gains all the advantages that a gambit could offer him... superior development, superior center, gain of time by chasing the exposed Bishop... without the need to gambit any material; while Black gains nothing in return.

You STILL haven't offered any supporting lines or ideas for Black in this variation.

A missing b-pawn gives increased activity at the expense of central control.

Having the b-pawn still present gives reduced activity than the Evans gambit, albeit with increased central control.

Both wins time.

You seem to be saying that "all other things being equal", the two lines are fairly even. But all other things aren't equal.

Your analysis ignores the fact that WHITE IS A PAWN DOWN in one of those lines, but not in the other. The game doesn't end in the opening (unless Black messes up completely).

Greco attacked prematurely instead of fully developing his pieces yet was still able to take advantage of his opponents having a poor defense, does a poor defense always require worse development than your opponents, considering that Greco still attacked prematurely and succeeded due to his opponent having bad defenses yet greco failed to develop properly.

Does the activity advantage remain well into the middlegame?

In the first place, Greco's play is known only from the games that he published in his manuscripts. In other words, we only see the games that he decided to preserve in writing... which means that we only see what he wanted us to see.

In the second place, Greco's games (in the manuscripts) are widely suspected to be compositions, rather than actual play against real opponents.

In the third place, he was so much stronger than his contemporaries that he could probably have beaten them regardless of what he played.

In answer to your question... no, a poor defense does not always require inferior development. Development is only one factor, and is only really important in open positions. A bad defense MIGHT involve poor development, or it might involve an organic weakness (a backward or isolated Pawn, for example), or unnecessary loss of material, or some other factor.

- Does the activity advantage remain well into the middlegame?

The missing b-Pawn only gives increased activity to the Bishop as long as it remains on its home square. As soon as it moves, the missing b-Pawn no longer benefits the Bishop at all.

The open b-file for the Rook? Who puts their Rook on b1 in the Evans? Dispersing your pressure all across the board is a good way to lose your advantage completely. Pressure should be focused on important points.

Is 1.e4 e5 2.nc3 nf6 3bc4 bb4 4.c3 open or closed?

The opening hasn't yet resolved into a position that can be considered either open or closed.

But the general rule is that superior development is increasingly important the more open the position is. In a position that has open lines for both Bishops and Rooks, development is quite important. In a blocked position (for example, a King's Indian Defense where the center Pawns are locked), superior development might mean very little.

Is black able to keep the position closed?

It's a lot harder to do so after 3. ... Bb4.

Consider this line:

So in other words the line discussed in this thread is bad for black because black is unable to keep the position closed and instead ends up in an open position where white is superior in development.

3. ... Bb4 commits Black to losing time (after 4. c3) which wouldn't be all that serious if the position could be kept closed.

But you still haven't offered any positive reasons for playing 3. ... Bb4. Why accept the loss of time if you gain nothing in return (compared to moves like Bc5 or Be7 instead of Bb4).

If you don't like the Fried Liver, then 3. ... Be7 is a perfectly acceptable alternative for Black.

What is the best way to take advantage of having superior development in an open position?

Generally speaking, you need a superiority of two pieces to launch a combination. So if the opponent has one defender in the target area, you would need three attackers. This is just a rough rule of thumb, but it usually works. Then you look for tactics... pins, forks, skewers, overloads, decoying and diverting operations, etc.

A few examples:

From an over-the-board tournament in the Canadian Maritimes:

Another over-the-board game: