Well, it is an idea. I doubt Illescas is meaning a clear advantage for white in move 5, neither he is trying to refute the Budapest. But even if so, I'd take his word over all that machine gibberish anytime.
How to play against Budapest Gambit
A repeat of the experiment.
SF does not have any problems to hold with black again.
I guess white retains a smallish edge after 4. e3 too, but that is a much less ambitious and weaker line than the Alekhine Variation.
All those lines that Pulpofeira posted don't give any real advantage to white.
SF around the 0.0 margin all the time after 4. e3 and Nh3.
Of course, the Budapest could be promising, if white plays weak.
Once again you immediately run to engine. All engine with you fraud.
No bigger fraud than you.
I should have reported you, but too tired now, tomorrow.
You might have tried to win that one by playing 33. Kc3 and Kb4 instead of f4.
SF does not see it, I have to suggest it to it.
Still not certain this is won for white, it is much more complex than the position we discussed in the other thread, but a nice try to win.
People say I am relying on engines.
How do I rely, when SF sees only draw everywhere here?
My intuition, based on countless number of games seen, suggests to me white is better and has good winning chances.

You might have tried to win that one by playing 33. Kc3 and Kb4 instead of f4.
SF does not see it, I have to suggest it to it.
Wrong forum --> Endgame Study.

Once again you immediately run to engine. All engine with you fraud.
No bigger fraud than you.
I should have reported you, but too tired now, tomorrow.
It would be mauvais ton to "report" Borat for such an anodyne remark.
All those lines that Pulpofeira posted don't give any real advantage to white.
SF around the 0.0 margin all the time after 4. e3 and Nh3.
Of course, the Budapest could be promising, if white plays weak.
Once again you immediately run to engine. All engine with you fraud.
No bigger fraud than you.
I should have reported you, but too tired now, tomorrow.
I not fraud. I not try to sell shitty book.
The only sh*t and fraud here is you.
I wonder why the mods still have not banned you?
People, this is Brendan J. Norman, the biggest insinuator in all of today's world.
Once again you immediately run to engine. All engine with you fraud.
No bigger fraud than you.
I should have reported you, but too tired now, tomorrow.
It would be mauvais ton to "report" Borat for such an anodyne remark.
Pas mauvais, ce serait un tres tres bon ton.
Anodin pur lui, pas pour les normes de moralites.

That's true for almost every opening.
The Budapest gives White a choice between a slight positional edge and a complex counter attacking duel. I prefer White, but the real problem with the Budapest is if White goes for the positional edge, Black doesn't have much chance to win against a strong opponent.
So if Black is in a must-win situation, there are better options and if Black is in a must draw situation there are better options.
Objectively tho, the Budapest shouldn't lose.
Illustrative game 2. The type of game you can face in some Budapest variations: