There actually is an established way to learn languages through cognates and etymology, given that many modern languages have been immensely influenced by classical languages.
- Classical Chinese
I am not familiar with the pedagogy involving most of the aforementioned tongues, but I am aware of two main methods, one of which is well-known, and the other surprisingly inconspicuous.
Lingua Latina per se Illustrata (Ørberg, 1955) is a famous book in which many essential Latin words are included, and have their meaning elaborated through contextual clues. (un)Coincidentally, words familiar to English speakers abound, which should be the case with other Indo-European language speakers, since a substantial proportion of the lexicon in all these languages are cognate with their Latin equivalents.
The English “family” should be well acquainted with the Spanish “familia” due to their common Latin ancestor, “familia”.
In the Sinosphere, Chinese characters have been used extensively beyond the Sino-Tibetan Chinese language to create a sprachbund among Japanese, Korean and Vietnamese. If you were to learn Japanese kana, you’ll realise how Chinese characters, and their Middle Chinese sounds, were borrowed to be fitted into the corresponding Japanese sounds. Yet this is seldom highlighted even to students who can read Chinese characters, who learn the kana through memorising the romaji transliterations. Of course, the more well-known aspect of this sprachbund is the Sinoxenic vocabulary common in all the languages.