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There is "coaster brake" mechanism, it consists of not only brake, but also of freewheel. As can be seen in answers in Why there are no or few bicycles with coaster brake and external speed gears? , coaster brake is incompatible with derailleur, because it "requires significant tension on the lower part of the chain when it is applied" while derailleur requires that the lower part of chain is relaxed. The lower part of the chain is tensioned only when pedal is pressed in direction opposite to usual ie, when the brake is pressed. (The usual direction is when pedal tries to accelerate bicycle forward.) So, seems, the problem appears only when the brake is applied. So, brake component of the "coaster brake" is incompatible with derailleur. (This means that pedal brake cannot be installed together with derailleur.) But what about the freewheel component of the "coaster brake"? Can such freewheel mechanism be used on the same wheel together with derailleur? I think it would be a viable thing, because it is noiseless. "Freehub" freewheel mechanism, which uses ratchet, is used nowadays with derailleur, and ratchet produces trilling noise while coasting.

I have seen a video of how coaster brake works,

. I feel from the video that the freewheel component of the "coaster brake" can be engineered and produced without the brake component.

Seems, derailleur is most popular gear change mechanism, nowadays. "Coaster brake" is also a popular mechanism, and seems it was relatively even more popular in the past. As I see, despite they both are so popular, the freewheel of "coaster brake" is not adopted for usage with derailleur for some reason, I have never seen such combination of bicycle components. Maybe somebody has tried, thinked about it, but found it impossible?

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    Criggie and I have both answered the question of "why not have a derailleur with a coaster brake". Best practice for Stack Exchange etiquette would be to narrow down this question to that subject, and then ask a new question about the noise made by mechanisms while coasting.
    – RLH
    Commented Aug 25, 2022 at 20:51
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    Your question had multiple components, and the presumption here is that the title provides the most accurate summary of what you had in mind when you asked the question. (Note that two experienced users both interpreted your question the same way). The answers also address the objective technical question from the original post, rather than the subjective fashion question. There's no reason to feel guilty about editing down a multipart question to match the part that other users answered.
    – RLH
    Commented Aug 25, 2022 at 21:00
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    @qdinar welcome to the site. You have the cross-site rep bonus, so clearly you know how SE works. These comments are not antagonistic, they're working to improve and clarify the question. If your intent is to ask about quiet freehubs, then why mention coaster brakes at all?
    – Criggie ♦
    Commented Aug 25, 2022 at 22:27
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    Possible duplicate of bicycles.stackexchange.com/questions/14194/… and bicycles.stackexchange.com/questions/40486/… and bicycles.stackexchange.com/questions/4606/… and bicycles.stackexchange.com/questions/15321/… and many entries in bicycles.stackexchange.com/search?q=quiet+freehub or at least relevant.
    – Criggie ♦
    Commented Aug 25, 2022 at 22:28
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    If the question really "why aren't silent freewheels popular", why is there a wall of text about coaster brakes, Soviet Russia and other unrelated things?
    – ojs
    Commented Aug 26, 2022 at 6:58

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Coaster brakes won’t work with external (i.e. derailleur) gearing.

Coaster brakes require the bottom section of the chain to be in tension (to pull against the brake assembly). Derailleurs need the bottom section of the chain to be effectively slack (so that the derailleur can move to take up extra chain length when in smaller-radius gears).

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