That's what's so baffling to me. If you read any history, e.g. the one I linked, you can also try the numerous ones on marxists.org, even wikipedia, honestly pick your source on any part of the political spectrum, you find out that there was a student movement, followed by mass protests, strikes at factories followed by a general strike, the Red Army was kicked out, a provisional worker's council committee was established that was democratically elected by workers, then the Red Army came back in force and ended it, after which there were more strikes and mass mobilizations that were repressed by the Soviet military and the Kádár government. There are many accounts of revolutionaries clarifying that they didn't want to return to capitalism, they just wanted the Red Army out so that they could build what they referred to as real socialism.
I'm well aware that Radio Free Europe was propagandizing throughout the 1950s, there were leaflet drops, and the CIA document shows a relationship with Béla Király after the revolution, when they sponsored his organization. The Artemov book on page 36-38 talks about the information warfare waged on Hungary by the West but it doesn't give an account of what actually happened.
The idea that Béla Király was training militants in the mountains with the CIA is also contradicted by the fact that he was imprisonned from 1951 until a few weeks before the revolution. As a sidenote, while he fought on the Axis side, he was later named one of the "Righteous Among Nations" by the World Holocaust Remembrance Center because of his relatively humane treatment of Jewish prisoners.
Anyway I just don't see the weight of historical evidence on the side of "reactionary coup", but I guess I should thank you for forcing me to do more research on this. I did learn a lot.