I'm not sure your implication holds. Let's look at your examples
In Texas's governor race, Greg Abbott won the nomination with 66% of the vote despite being labeled a RINO.
Greg Abbott is both an incumbent and a Republican in a deep Republican state. While he had several challengers, there was never much movement in the non-Abbott vote. Abbott briefly was underwater last year (50-41 disapproval polling), but has been steadily trending back up. Also, it seems like most of those calling him a RINO (Republican In Name Only) were his opponents in the primary.
In Georgia, Jody Hice and David Perdue lost their challenges, with Perdue losing by a stinging margin.
Both of these challenges were spurred on directly on directly by Donald Trump, who openly wanted political revenge for neither Gov. Kemp, or SOS Raffensperger supporting Trump's narrative about the 2020 election being stolen.
The former president has repeatedly attacked Raffensperger, along with Georgia Republican Gov. Brian Kemp, for rebuffing his efforts to overturn President Joe Biden’s victory in the Peach State in the 2020 presidential election. Both Raffensperger and Kemp notched convincing wins Tuesday even though they drew the former president’s ire, according to NBC.
In North Carolina, an incumbent America First candidate, Madison Cawthorn, lost his primary presumably because of scandals. No incumbent Progressives came close to losing primaries that I'm aware of.
Cawthorn has been embroiled in several scandals
Cawthorn’s electoral prospects had rested on sandy soil for weeks before the election. There were the multiple tickets for speeding and charges of driving with a revoked license, the times he tried to carry a gun onto an airplane, and his claim that he had been invited to an orgy by his colleagues in Washington. Shortly after, there were the photos of him in women’s lingerie; the video of him simulating sex with another man, whom he has referred to as his cousin; the allegations of insider trading and improper relations with one of his staffers; and the time he called Ukrainian President Volodymyr Zelensky a “thug.” Kevin McCarthy told Cawthorn that he had lost the House Minority leader’s trust. (Cawthorn declined my request for an interview.)
Such losses are rare
In one sense, he lost narrowly, finishing second to Edwards by about thirteen hundred votes, out of nearly ninety thousand cast. By another measure, the vote was a rare repudiation of a sitting member of Congress. (Over the past half century, more than nine in ten House incumbents have won reëlection.) In his race, sixty-eight per cent of voters chose one of the seven candidates not named Cawthorn.
In the 2022 primaries, in terms of high profile races, every progressive candidate, except for Nina Turner, did not suffer huge losses like America First candidates are so far. Summer Lee, a Progressive narrowly won her primary, and it looks like Jessica Cisneros might as well in a runoff though it seems more likely she narrowly loses.
In general, trying to unseat an incumbent who is not unpopular is fairly hard in both parties. As such, it is less common to see serious primary challenges from within the party. What's made 2022 different is that Donald Trump has inspired a large number of people to make more serious bids against incumbents (with or without any explicit endorsement from him). Conversely, Trump has backed a large number of incumbents who were polling well allowing him to claim his endorsements are meaningful.
But Trump is also promoting more individuals who are aligning with his values who cannot win elections. That is why more primary challengers on the Right are losing.