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https://en.wikipedia.org/wiki/Thought-terminating_clich%C3%A9
As the party fielded more and more candidates who championed economic-right policies in order to attract corporate donations, the idea of "purity testing" was turned into a stock phrase to attack critics using labels, instead of justifying it on its own terms.
Phrases like "moral purity" now serve two functions: firstly, to dismiss criticisms from the Left wholesale without having to discuss them directly, and secondly, to blame them for the rise of fascism.
In theory it was also supposed to serve a third function of bullying the Left into voting Democrat, but that didn't work.
What are you on about? There is no progressive wing. The center doesn't need the left to get anything done, as evidenced by the last 50+ years. All they need is the permission of the right.
Again, you're complaining that there's no space for your platform but that has nothing to do with centrists secretly favoring the right. In the USA (and most 21st century democracies, to some extent) we have a rigged spectrum with only Center, Center-Right and Hard Right.
If you want to play the political game as it stands then you're submitting yourself to the Center. It's not back stabbing, you just have no political weight to merit anything but lip service. Gaining the clout to influence policy isn't going to come from the good will of the Centrists, it has to be built independently.
It's arguably an insurmountable task but it starts by making policy that appeals to voters across the current spectrum. That means a hard focus on economic solutions for the wage-earning class. Look at the success of Bernie, AOC and Mamdani, their focus is generally-to-exclusively economic. Once you build that Center-Left you sap some life from the right and can build from there.
Of course, all of this is going against coordinated establishment attacks. You can see those headliners getting it from all media outlets, but as long as they stay on message they have success. Attacking them for not being progressive enough is silly when they're actively pushing the limits of what the system allows. That is the essence of the "purity test".