Learn English – I’m looking for a word, or a phrase that would describe someone’s outlook and position

One who doesn't take a stance on a subject can be characterized as impartial or indifferent

impartial (adj) - treating all rivals or disputants equally; fair and just:

indifferent (adj) - having no particular interest or sympathy; unconcerned*

One who can't make decisions on his own is certainly diffident:

adj - lacking confidence in one's own ability, worth, or fitness; timid; shy.

There are many options though. You should be more specific and maybe provide some context: apathetic, infirm, weak-willed, detached, aloof are all terms that spring to mind.

Medically speaking, a person who can't make his own decisions and constantly depends on others suffers from Dependent Personality Disorder.

New Oxford American Dictionary*

A few similar phrases are common but they don't quite perfectly match. The most obvious is "informal":

informal — free of ceremony: relaxed and casual rather than ceremonious and stiff

This doesn't inherently imply anything about status or position but it can be used to describe a scenario or situation where formalities such as specific procedure involving treatment of persons with such status or rank are not necessary. Pragmatically, however, persons of lower status would still treat those with higher status with greater respect.

The phrases "speak freely" or "act freely" are very commonly used in American fiction to allow behavior between ranks regardless of procedure or expected respect. The stereotypical example is used in strict hierarchies such as the military:

Alice: Permission to speak freely, sir.

Bob: Permission granted.

Alice: I think your idea is terrible and one of the worst suggestions I've ever heard.

This phrase does not, however, apply to entire situations such as a meeting. You could use either phrase to describe the effect:

We felt the ability to act freely, regardless of status.

But it doesn't quite have the compact, self-contained meaning associated with the Japanese or Chinese words you have defined.

Aside from those two, I can only think of overly specific terms such as "double-blind" or "brainstorming session". These two have similarities at heart but would never be used in the way your Japanese or Chinese words have been defined.

That being said, if your specific ask is this:

Apart from the provenance of “Bureiko” and “bù fēn zuò cì”, is there any English short word corresponding to “Bureiko,” or “Bù fēn zuò cì” in the meaning of equal opportunity of speech and opinion among all?

There is a concept in logic that specific notes that the origin of an argument is completely irrelevant as long as the logic is sound. This concept is most often expressed through its inverse: A logical fallacy known as the "appeal to authority":

Argument from authority (Latin: argumentum ab auctoritate), also authoritative argument and appeal to authority, is a common form of argument which leads to a logical fallacy when misused.

The basic form of the fallacy is assuming that a particular argument is more correct because it is associated with some authority on the subject. The most flagrant forms of this fallacy will borrow an authority from an unrelated topic (e.g., a sports star) to make a positive claim about a particular product (e.g., an energy drink).

Completely rejecting this fallacy would rely on completely ignoring all status and position when listening to ideas or arguments.

Again, this is not a very accurate translation of your word. It merely holds a few similar concepts.