It must be remembered that in the Japanese language system, the lexeme's sound and the lexeme's spelling are much less correlated with each other than even in Chinese; the phenomenon of 訓読み kun'yomi means that one lexeme can be written with several different spellings (even if the spellings overlap in meaning). The most well known example for beginners is あう = 会う or 合う or 遭う etc. which all have slightly different connotations on paper for "to meet".
Although かわいい has only one kanji spelling 可愛い, it is known to be an ateji 当て字, i.e. neither 訓読み kun'yomi nor 音読み on'yomi. Hence the etymology of the かわいい is probably derived from something else, despite the similarity of the 音読み on'yomi of 可 (か) and 愛 (あい) to かわい.
Wiktionary has an alternative to the straightforward "Classical Chinese to Japanese loan" etymology, one that is borne out by 語源由来辞典 (Gogen Yurai Jiten)'s entry. According to this, the original expression was 顔映ゆし (historical kana かほ kaho + はゆし hayushi), attested from the late Heian period, originally meaning "face radiating" in a negative sense, i.e. embarassed or shy.
The assumed phonological development of 顔映ゆし goes as: kahohayushi > kawohayushi > kawahayushi > kawawayushi / kahahayushi > kahayushi > kawayui, which is one possible reading of 可愛い. This regularly becomes かわいい kawaii.
One of Wiktionary's quoted sources is the Dictionarium sive thesauri linguae Iaponicae compendium compiled by the Spanish-born early 17th century Dominican missionary Diego Collado. Under the entry for "miserabilis", i.e. pitiful (emphasis mine):
Miserabilis, e: cosa que causa lastima, cavaij.
Hence by 1632, the pronunciation had shifted to かわいい although the meaning had only shifted a little, from embarassed to pitiful. From there, the shift to adorable and cute is quite straightforward but it is also something that has occurred in the last 400 years.
The written collocation of 可愛 has a much longer history in Chinese, dating back to the 書經 Shūjīng / 尚書 Shàngshū (the "Classic of Documents"), traditionally dated to the late Zhou, where it exists with a rather transparent original meaning in Chinese. In chapter 3, the 大禹謨 Dà Yǔ mó "Counsels of Great Yu", the following question is posed:
可愛非君？可畏非民？
Of all who are to be loved, is not the ruler the chief? Of all who are to be feared, are not the people the chief?
This is certainly not what we in the 21st century would call cute or adorable; again, in fact like in English, there has been a semantic shift of "that which is to be adored and loved" to "cute".
So even if they share the characters, it is only the modern use that has become aligned; the roots of the words themselves are rather further removed from each other.
