One picture holds this whole family together: something detaches and drops. A leaf leaves the tree, hair leaves the skin, a pupil leaves school. The verb is the bare act of falling off; the noun נְשִׁירָה names that falling as a process; and the soaring nesher sits here too, the great bird linked to the same letters. Every word below 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.
This root is mostly intransitive: things fall off by themselves. Hebrew never lexicalized a causative hif'il or a passive partner here, so the whole burden of meaning sits in the simple pa'al plus a few nouns built straight off it.
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.