I don't care if people dislike this - to reject something properly involves to know it.
Matthew 19:42 - - ποτίσῃ ἕνα τῶν μικρῶν τούτων ποτήριον ψυχροῦ μόνον εἰς ὄνομα μαθητοῦ
The Petelia Gold Tablet
δίψηι δ’ εἰμὶ αὔη καὶ ἀπόλλυμαι. ἀλλὰ δότ’ αἶψα
ψυχρὸν ὕδωρ προρέον τῆϛ Μνημοσύνηϛ ἀπὸ λίμνηϛ.’
I am parched with thirst and dying: quickly, give me the cool water
flowing from Memory’s lake.’
So that's a gift of cold water as one of the symbolic stages in entering the afterlife in a mystery religion of pupils and teachers.
Robert Price discusses early Christianity's situation relative to the Graeco-Roman mystery religions in the history-of-ideas at length in https://archive.org/details/DeconstructingJesus/page/n159/mode/2up
The image had entered ancient education via Plato's Republic, and was more likely to have been familiar from popular usage than to be Jesus quoting the Petelia Gold Tablet, or the general sort of thing it was (which we don't know)
That suggests a reading where to drink the cold water is to remember.
And the gift of cold water is an informational gift - perhaps quite akin to our modern popular image of 'the red pill and the blue pill' which always refers to the film the Matrix and never to Sigmund Freud's pleasure principle and reality principle.
It doesn't suggest it very strongly, but this is the sort of thing that should be noted in case other analogies of spiritual drinking are found to relate to discourse, or memory, etc.
The word ψυχρὸϛ has a wide register for what a simple sensation it describes.
It's for cold things that etymologically have been blown on, relating it to ψυχή the soul or breath. There's a very close cognate ψυχὸϛ which gets a slightly different meaning in the lexicon (but that's probably just because lexicons of ancient languages are terrifyingly incomplete). There is also a verb ψυχω which combines blowing to cool with a wide range of other things like making something dry, or emotional coldness (Matthew 24:12), but not life being breathed into Adam (ἐνεφύσησεν) or the inspiration of the holy scripture (θεόπνευστος).
At Jeremiah 6:7 a well keeps its water cool "ψύχει λάκκος ὕδωρ" (and thus Jerusalem keeps her evil fresh), which might be important to the OP's point.
I find the inference from history appealing, but neither the simplex image Jesus describes nor its converses require the water to come from a clay jar, or imply that it has done, etc.
There is probably something else going on.
We might take the "only", together with the emotional coldness of Matthew 24:12, and together with some historic stranger-guest etiquette or desert cultural norm that giving a child water is the least someone can do. There is apparently a term for this: hachnasat orchim with modern customs that have dated back to antiquity, but I can only suggest it for further study.
The choice of ψυχρὸς rather than κρύος - which was probably more common and less confusible with ψυχή - probably is because we're in an analogy about someone coming to Jesus for salvation. There isn't enough to derive other information about the intended link though - beyond the obvious that it's vital and a gift.