One idea threads this whole family: taking a lump and breaking it into single units. Break a banknote into coins, break a topic into points, break a state-owned firm into private hands, even pluck single strings on a guitar. From that act come the noun for one detail (and one individual), the adjective private, the itemized breakdown, the menu of separate dishes, and the word for privatization. Same three letters, seen from the angle of 'the single part'. Every word is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
Watch the three binyanim split the idea: pa'al physically breaks the whole (cash, strings); pi'el breaks a topic into listed details; hif'il breaks public property into private parts. Each binyan applies the same 'break into singles' move to a different field. The passive partners are not lexicalized for daily use.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.