I'm trying to remember a story, which (given the content) was almost certainly written in the 1950s.
"The Rocket of 1955", a 1939 short story by C. M. Kornbluth. It was reprinted in Worlds Beyond, February 1951, which is available at the Internet Archive. Any of these covers look familiar?
It features a private manned rocketry program, which turns out to actually be a murderous scam.
The requested contributions poured in. Newspapers and magazines ostentatiously donated yard-long checks of a few thousand dollars; the government gave a welcome half-million; heavy sugar came from the "Rocket Contribution Week" held in the nation's public schools; but independent contributions were the largest. We cleared seven million dollars, and then started to build the space-ship.
The virginium that took up most of the money was tin-plate; the monoatomic fluorine that gave us our terrific speed was hydrogen.
The people building the supposed spacecraft actually produce a dummy, which explodes right after takeoff.
The vaned projectile roared a hundred yards into the air with a wobbling curve—then exploded.
I remember a bit about the crew being sealed in, just as they realize that the controls they have been practicing on were not there.
The take-off was a party for the newsreels: the big, gleaming bullet extravagant with vanes and projections; speeches by the professor; Farley, who was to fly it to Mars, grinning into the cameras. He climbed an outside ladder to the nose of the thing, then dropped into the steering compartment. I screwed down the sound-proof door, smiling as he hammered to be let out. To his surprise, there was no duplicate of the elaborate dummy controls he had been practicing on for the past few weeks.
And the villains would have gotten away with it all, except that one of them got drunk and blabbed.
But the professor got drunk and, disgusted with the part he had played in the affair, told all and poisoned himself.
I think it was the first story in an old anthology I had found at my grandparents' house.
Reprinted in many anthologies and collections, it was indeed the first story in Kornbluth's 1976 collection The Best of C. M. Kornbluth.