One feeling sits under this whole family: being able to rely on something. From it grows a chain of meanings: to trust (pa'al), to make something reliable for someone, to promise or ensure (hif'il), and to put a money-backed safety net under a risk, to insure (pi'el). The nouns then name the states of safety: certainty, a refuge, security, and the abstract bitachon that means both inner confidence and national defense. 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.
Three binyanim, three flavours of reliance: pa'al is the inner act of trusting, hif'il makes a thing reliable for someone (a promise, a guarantee), and pi'el specializes into the financial sense, to insure. The nif'al and hitpa'el slots are not lexicalized for this root.
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.