I’ll start by giving the passage on this change from Elementary Middle English grammar : James Wright, as a change specific to French loans.
§231. Initial e– disappeared before s + tenuis as Spaine, spȳen, staat beside estaat, stüdien, scāpen beside escāpen, squirel (O.Fr. escurel). Initial vowels also often disappeared before other consonants, as menden beside amenden, prentȳs beside aprentȳs, pistīl beside epistīl. Initial prefixes often disappeared, as steinen beside desteinen ‘to stain’, sport beside disport, saumple beside ensaumple.
Now, as OP points out, Latin > French and Latin > Italian went the other way.
The insertion of an initial e- before a cluster makes a word easier to pronounce; Latin statūs > Old French estat (Modern état) > English estate (state). Lots of languages do this; Turkish is another good example; French station > Turkish istasyon.
So why would English go the other way?
Notice that this change happened to French loans; it isn’t something that happened generically in the language, to Old English words. And ease of pronunciation is not an absolute in a language; after all, plenty of languages do have words starting with st-. Like Latin.
Or Old English.
If a change systematically happens to French words, it might not be motivated by making them easier to pronounce. It might be motivated by making them look more familiar—which can mean making them better aligned to the native phonotactics of the language.
So: did Old English have lots of words starting with an unaccented esc-, esp-, est-? From what I’m seeing at An Anglo-Saxon dictionary : based on the manuscript collections of the late Joseph Bosworth , no: just a couple of words starting with accented est-. As opposed to lots and lots of words starting with sc-, sp-, st-. And of course native stress was on the first syllable: an unstressed initial esc-, esp-, est– would have sounded doubly alien to Old English.
The rule was not regular and overwhelming: we’ve gone back to escape from scape, and we’ve kept apprentice. But—without knowing this for a fact—I think that’s what’s happened.