One idea runs through this whole family: being fit for purpose. The English borrowing "kosher" is just the most famous corner of it. Underneath, the root means qualified, capable, up to standard, whether the standard is religious law (kashrut), professional skill (qualifications), or a body that is in shape (fitness). The verbs let you make something fit, the nouns name the fitness, the certificate, and even the tool that does a job. Every word here 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.
The pa'al "to be fit" is rare in everyday speech; the working verb is the hif'il לְהַכְשִׁיר, the causative "to make fit", to train people, to qualify a tool, to make food kosher. Same act, three contexts.
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.