Why is the Budapest Defense considered not fully sound or even dubious? What theoretical difficulties arise for Black? [fen &quot;&quot;]
1. d4 Nf6 2. c4 e5 3. dxe5 Ng4 4. Bf4 (4. Nf3 Bc5 5. e3) (4. e4 Nxe5 5. f4) (4. e3 Nxe5 5. Nh3)

White can choose from multiple setups, most common are 4. Bf4, 4. Nf3, 4. e4 or even 4. e3 with the deep idea of Nh3-f4-d5. Database results favor White heavily. Then again, legends with the caliber of Gelfand or Kramnik go down against it.
From what I gathered, people claim that Black has long-term strategical deficiencies, in many variations White can obtain the bishop pair and the positions are just &quot;good for White&quot;. But what exactly are the problems Black faces when playing the Budapest?

EDIT: I have found an interesting line in the Adler-Maroczy Variation (4. Nf3), where I think Black does not have enough compensation. Surprisingly, it is not analysed in any Budapest book I own and played extremely rarely, even though this is the move White wants to play according to most analysis (why is it not checked then???) and it is almost entirely forced.
[fen &quot;&quot;]
1. d4 Nf6 2. c4 e5 3. dxe5 Ng4 4. Nf3 Bc5 5. e3 Nc6 6. Nc3 O-O (6...Ngxe5? 7. Nxe5 Nxe5 8. f4! Ng6 9. Bd3 {is regarded as very good for White and is warned against in almost all Budapest analysis}) 7. Bd3! (7. Be2 Ngxe5 {is normal and Black is fine}) Ngxe5 (7...Re8 8. Qc2 {Black has to somehow block the attack on h7, either by ...g6 or ...h6, both disabling the vital Black plan of Re8-e6-h6 with a kingside attack. White has ideas of returning the pawn on his terms with e6} (8. Bc2 {White achieves a good placement for his bishop without problems})) 8. Bxh7 Kxh7 9. Nxe5 Nxe5 10. Qh5+ Kg8 11. Qxe5 b6 {Black is a pawn down, in return has the bishop pair, but will probably have to forfeit it soon, as ...Bb7 will be blocked by Nd5. White can castle both sides after Bd2}
It's definiely not a &quot;bad&quot; option for Black. It can be played at high levels and still get decent results. However, it's not considered a &quot;great&quot; line for many reasons:

As you said, White has multiple set-ups to choose from most of them leaving them with a slight advantage.
The &quot;surprise factor&quot; that may give Black with an edge against an unprepared opponent doesn't exist (at least not so early in the opening) at the highest levels.
White only enters the Budapest if he wants. If he's worried that his opponent may go for it, he can just play 2.Nf3, most likely transposing later on to his favourite opening.

Database statistics should always be taken with a grain of salt, since there could be multiple factors biasing it (a particular player that uses it very often, or an opening being &quot;the favourite&quot; of weak players, or maybe an opening that is used as a surprise weapon against strong opposition). However I think the factors above prevent the Budapest from being a practical choice to build your repertoire around (specially reason #3: you'll still need something against 2.Nf3, say the King's Indian. You'll be able to use that against 2.c4 too, so you may just as well forget about the Budapest and focus on the King's Indian)