Where does “cooler” come from?

I notice this etymology chain appears to have internal inconsistencies and repetitions (Proto-Indo-European gel- appears twice, Latin -are appears twice, and the chain seems to loop back on itself). Additionally, the chain from English "cool" to Chinese "兒" to Proto-Indo-European doesn't represent standard etymological relationships. I cannot write an accurate one-sentence origin summary for "cooler" based on this chain, as doing so would violate the instruction to "state only what the chain supports" and "do not invent." The chain itself appears malformed. Could you provide a corrected etymology chain? A typical path would trace English cooler → English cool → Old English col- → Proto-Germanic → Proto-Indo-European, or similar legitimate relationships.

cooler (English): Anything which cools; An insulated bin or box...

Definitions

  1. Anything which cools; An insulated bin or box...

Ancestry of “cooler”, step by step

cooler traces back through two separate lines of ancestry.

via English cool

StepLanguageWordMeaning
1EnglishcoolHaving a slightly low temperature; mildly or...
2Middle Englishcool
3Middle Englishpekpeak
4Middle EnglishpikeAlternative form of pyke; pike; sharp point, iron...
5Old Englishpīcpointed object, pick axe
6Proto-Germanic*pīk
7Latinpixpitch, tar
8Proto-Indo-Europeanpik-to be angry; pitch; resin
9Proto-Indo-Europeanpi-sap, juice

via German Kühler

StepLanguageWordMeaning
1GermanKühlercooler; radiator
2German-erForms agent nouns etc. from verbs, suffixed to...
3Old High German-āriused to form agent nouns
4Proto-West Germanic-ārī-er
5Proto-Germanic-ārijaz-er
6Latin-āriuser
7Proto-Italic*-āziosForms relational adjectives to nouns (and rarely numerals)

Words derived from “cooler

Every word from Proto-Indo-European pi-Every word from Proto-Indo-European pik-Every word from Latin pix