I believe the raising Sardaukar-style force by Leto, as mentioned by Thufir, would be the reason Corrino turned on Atreides.
True, Leto was a popular man, was one of Imperial Kinsman (actual, not just titular), with, apparently, strong claim to the throne, but all that was politics. Serious, complicated, but not really anything outrageously exceptional. Landsraad was, by definition, a counter to Corrino's power:
"You know as well as I do what forces surround us. We’ve a three-point civilization: the Imperial Household balanced against the Federated Great Houses of the Landsraad, and between them, the Guild with its damnable monopoly on interstellar transport. In politics, the tripod is the most unstable of all structures. It’d be bad enough without the complication of a feudal trade culture which turns its back on most science.”
So rising to prominence in the Landsraad is inevitably antagonistic to the Emperor, even though Shaddam had his own support there, too. Can't imagine this never happening before...
House Corrino rose to be Emperors almost exclusively due to their military power, which were Sardaukar.
Furthermore, the fact that Harkonnens didn't know why Corrino wanted Atreides gone is a strong indicator that it wasn't politics, it was challenging the Sardaukar. Considering what Leto said to Paul:
"They wish the Atreides name to become unpopular,” the Duke said. “Think
of the Landsraad Houses that look to me for a certain amount of leadership—
their unofficial spokesman. Think how they’d react if I were responsible for a
serious reduction in their income. After all, one’s own profits come first. The
Great Convention be damned! You can’t let someone pauperize you!” A harsh
smile twisted the Duke’s mouth. “They’d look the other way no matter what was
done to me."
It is quite obvious that dealing with Atreides in Landsraad in manner described above would be faster, cheaper, surer and less dangerous to balance of power. Especially since Shaddam IV had no heir, so Corrino would no longer have claim to Emperor's throne. So there was no legitimate reason to fear any Leto's attempt at it.
No, the real threat had to be to the powerbase of the Corrino, because this is the only thing that justifies risk of instant rebellion, had word of truth about Arrakis Affair got out. Because what was done to Atreides was one thing that everyone in the Landsraad was truly afraid of: being picked by Corrino one by one.