χ Marks the Spot
When Latin borrowed Greek words containing the letters χ, φ, and θ, they were rendered as ch, ph, and th, indicating their original aspiration. This is still the way most folks transcribe Ancient or Koiné Greek. Thus σχολή (schole) becomes Latin schola, though scola also appears.
Old English scōl ignored this particular nicety, as did, for the most part, Middle English:
Nu hauede þe king Aruiragus enne sune..he wes isende to Rome to leornien in scole.
Now King Arviragus had a son … he was sent to Rome to be educated in school.
The Early English Books Online I corpus shows the current spelling emerging in the later 16th c.:
So am I in good hope of your like affection, zeale and good vvill, for and in the erecting, and establishing of a free gramer school vvith in this citie … — John Hooker, Orders Enacted for Orphans, 1575.
This concordance betwene the parent at home, and the teacher in school for the vertewous training vp of their litle young ones, is in verie dede, to bring them vnto Christ, … Richard Mulcaster, The first part of the elementarie vvhich entreateth chefelie of the right writing of our English tung, 1582.
And a spelling with no h, though with the double vowel, still in the late 17th:
The greatest courage that ever yet rul'd, was baffled by fortune, tho' ne're so well scool'd, … John Playford, The Second Book of the Pleasant Musical Companion, 1686. EEBO
3: scool pastime for children, 5: the Countrey schoolmaster — John Newton, The Compleat Arithmetician, 1691. EEBO
So say I, he might be better employed in teaching scool, or any other honest occupation, then in thus Villifying and abusing an honest people … Edward Penington, A Modest Detection of George Keith's (miscalled) Just Vindication, 1696.
These represent a small number of orthographic holdouts: In the EEBO I corpus, from 1640 to the end of the century, there are 9877 attestations for school and only 17 for scool. Over the first four decades of the 17th c., then, the Latin-inspired spelling supplanted the one closer to Middle English.
The Bigger Picture
This hardly took place in isolation. Already in the early 15th c., someone with a degree of fluency in Latin took a look at dette, from Old French dete, and realizing the word must come from Latin debitum, decided the English word needed a b, though it would never be pronounced:
Þan my vessells of siluer...be sold and paied for my debtes wher myn other gods faillen. — Wills in Bedfordshire, 1415.
The rage to make English words more transparent to their Latin or Greek origins gets in full swing during the English Renaissance. Receipt from ME receit, OF receite, adds a p to bring it line with feminine past participle recepta. In the etymological history of anchor, Lat. ancora, however, there never was an h, but it got one anyway from a corrupt Latin spelling.
Perhaps most ridiculous is the bird ptarmigan, from Gaelic tarmachan. The Greek word for ‘wing’ started with a p — think helico-pter or pterodactyl, lit. ‘wingfinger’ — thus Gaelic gets a Greek appendage in the 1680s. At least somewhere in school’s dark past, there was an h that made sense as a Greek relict.
Latin threw in a h when borrowing Greek words containing χ, φ, or θ to show aspiration. The rh in rhythm, rhyme and diarrhea/diarrhoea, however, indicates a strong trill in the original Greek, or, in the digestive disorder, an internal double rho that signaled the same thing. OE and ME — though a few ME manuscripts Latinize to schole— basically ignore this refinement peculiar to Latin, until for etymological reasons, writers in the later 16th c. glue it back in.