How is White compensated in these lines of the Milner-Barry gambit?

Question

After White's sixth move 6. Bd3 in the Milner-Barry gambit line, White hasn't yet truly offered his pawn up for sacrifice yet. It's for this reason that Black will play cxd4, but not continue the exchange following cxd4, instead fortifying against a possible Bb5+ with Bd7. What if Black chooses not to begin the exchange until he can be sure to win it? After all, is usually happy to return dxc5 with Bxc5, so there shouldn't be any rush to trade off the c-pawn. Yet when I follow the "best play" by Stockfish (and uniquely best, if it is to be trusted) following 6. ... Bd7 and arrive at the following position:

[FEN "r3k1nr/pp1b1ppp/1qn1p3/2bpP3/8/2PB1N2/PP3PPP/RNBQ1RK1 b kq - 1 8"]
1. e4 e6 2. d4 d5 3. e5 c5 4. c3 Nc6 5. Nf3 Qb6 6. Bd3 Bd7 7. dxc5 Bxc5 8. O-O

I see White has traded away one center pawn, and the other center pawn needs to be defended by pieces for the foreseeable future, and he doesn't have the attacking chances that could be expected from the mainline. Following Black's likely 8. ... a5, White can't even chase away the Black bishop with b4. Despite this, Stockfish rates White at about 0 to +0.5. What resources does he have?

There is also the following similar line, where if I didn't know any better, I'd say White ends up with many of the same problems including a meddlesome knight on his kingside, yet Stockfish rates this as about equal:

[FEN "r1b1k2r/pp3ppp/1qn1p3/2bpP3/6n1/2PB1N2/PP2QPPP/RNB2RK1 b kq - 3 9"]
1. e4 e6 2. d4 d5 3. e5 c5 4. c3 Nc6 5. Nf3 Qb6 6. Bd3 Nh6 7. dxc5 Bxc5 8. O-O Ng4 9. Qe2

Answer 1

I am not too excited about this line for white either. Probably there is a reason that it is not played much.

What resources does he have?

As in most French, white has more space, two good bishops (vs one good for black) and if black castles short there are always ideas with attacking the king because most of black's forces are concentrated on the queen side.

Also, because you took on c5, white does not have to worry anymore about black taking on d4, making it easier to play particularly for a more positional player.

As for future plans, black most likely wants to play a5 to prevent white playing b4, Be3, a3, c4... Black's pieces are currently placed ok, but there is not much they can do from where they are. Likely sooner or later black wants to attack in the center with f6 or even d4.

After black's a5, the square b5 gets weak and white can target it developing for instance with a4, Na3, Nb5, perhaps later putting a knight on d4.

And if black castles short kingside attacks are always an option.