Etymology of close |kləʊz| (klōz) & close |kləʊs| (klōs)
Well, let's work from the basics on Etymonline. All senses of the word derive ultimately from Latin clausus, the past participle of the verb claudere. The verb form made its way to English from the past participle stem clos- of Old French clore /klor/, to shut or cut off from. The adjective form meaning strictly confined or secret is from the related Old French clos, and while /s/ is usually voiced in French, Old French had unvoiced final consonants.
So that was the original source of the difference, and it stuck and developed over time. The adjectival meaning of near is from the 15th century, by way of closing a gap.
They always had different pronunciations. Verb close (which the noun was derived from) used to have the "e" pronounced, which caused the voicing of the "s", while adjective close never did — the "e" was only to indicate the long vowel.
From glancing at the OED, this meaning appears to be used in the English Midlands and maybe Scotland (though to what extent I don't know). I think this would be unfamiliar to most Americans at least.
I'm not sure that it was ever particularly common, or that there is an answer for why it isn't used more.
Both of these words came from the same Latin root, but were borrowed into English at different times. Verb close came from Latin claud-ĕre via the Old French present subjunctive stem (clos-), first cited in 1275. The adjective close came from Latin claus-um, past participle of claud-ĕre, via French clos, first cited around 1400.