I hypothesize it has to do with isolation.
To get a peasant rebellion going, you need to isolate the peasants from their previous Pope. If you cut off their contact, they will functionally forget the old Pope exists in a matter of hours; by default you'll be the highest-status person in the room and they'll do what you say.
With the advent of newspapers and radio, the Pope can continually remind the peasants that he exists, so this cutoff cannot be accomplished. The peasant is expecting you to form your own FCC and ban the other Pope's radio. To successfully carry this out, even locally, you must have already militarily dislodged the old Pope; you have no need for a rebellion. America's habit of hiding its Pope makes this especially hard. Who are you supposed to overthrow, exactly? Bonus: when there's an internal usurpation, the peasants can't even tell that something happened, so their hero-worship of the system is not besmirched.
Elites are not stupid; they know this without me telling them, and as a result American elites don't even attempt it.
We can also see a historical parallel in Stalinism. Indeed Stalinism collapsed because they couldn't keep American radio out, and as a result the clearly higher-status American Pope trumped Stalin.
Of course peasants always prefer paleolithic Fascist sociology, but why not have Fascist sociology and be rich too?
I guess Putin's lucky the orthodox church doesn't have a Pope. The primitive radio-equivalent, churches, used to continually remind the peasants that there was a second master they might follow. To get a rebellion going you could make them believe you were more holy than the king using Sophistry; it's not like the Catholic Pope would notice in time, or indeed care overmuch even if he did. Bit of a downside to that whole [divine right] nonsense. The king felt he couldn't shut off the church/broadcasts, but you could easily interrupt the genuine line of control from the Pope long enough that Pope felt reversing fait accompli wasn't worth the effort.
In present Russia, TV and podcasts continually remind the Russian that Putin exists. Inertia is on his side; in the same way the king couldn't shut off the churches, you can't shut off their TV. (Not without fully indulging in cult indoctrination, which doesn't scale enough.) Meanwhile Russian churches only tell of an abstract master which peasants can't relate to. The Russian peasants are expecting you to seize Minsvyaz and ban Putin broadcasts.
Bonus: America's obvious slide into an unholiness spiral makes them seem lower-status than Putin, so he doesn't need to worry in the slightest about keeping American radio out. Yes America is Fascist and the peasants find Fascism terribly seductive, but between American psychosis and the educational effects of Communism, they grasp that Fascism is worse than Putinism.
The madness only works with intense peasant-psyche massaging from the direct Pope line. As long as Russians aren't glued to American TV, the process isn't intensive enough. If they lose concentration and go about their daily lives for a bit, America gets too fake and gay to take seriously by the time they return. Unless USG starts broadcasting in Russian, which it is far too narcissistic to do, Putin simply doesn't have to worry. I suppose it's also too late to try that; the catch-up process is even more intensive. Ironically radio is killing the radio Pope. The dependence of kratia addiction is always ultimately lethal, just like any hard drug.
P.S. China's peasants also expect you to ban the opponent's broadcasts, and the CCP dutifully carries this responsibility out. It's not that he minds being called Winnie the Pooh or anything, but using the words marks you as his enemy, and why would he give air to an overtly stated enemy? The Chinese, from top to bottom, can't see any reason.
American peasants largely can't see a reason either.
You give a couple symbolic sops and otherwise crack down as cracky as you can manage. If it doesn't sound like stone shattering you're not oppressive enough yet.
P.P.S. As king Henry found out, in reality local military superiority lets you boot the Pope whenever you want.
Nothing says "holy" to a peasant quite like
ad baculum. Peasants are clearly amazing, and whichever peasant is thinking this is the most amazing; obviously nobody would be able to beat him up without God granting them a divine exosuit to double their punching power, right? Right. They don't follow the king because they're scared shitless of their own shadow, they follow the king because God is clearly telling them to demonstrate grace by graciously allowing this inferior being, this pompous insect, to pretend to be in charge...