Cognates and illustrations definitely assist, and the book uses both freely. Chapter 1 begins with a map, for example. It’s a very clever strategy, one especially well suited for teaching Latin to people already speaking an Indo-European language.
I imagine it would be less useful to people who speak a language that doesn’t share the features of Latin. A map isn’t going to be of much use if one’s native language doesn’t have the concept of plurals, and one therefore struggles to comprehend the est / sunt distinction altogether. Then again, English doesn’t decline nouns or conjugate verbs (for the most part), and English speakers tend to be big fans of LLPSI. To its credit, it goes slow, and reinforces its lessons very well. The rest of chapter 1 is basically spent on variations of the above grammar point, introducing new nouns but reusing those two verb forms again and again in different variations, illustrating their use and how they contrast.
To your question, I also wonder how well this kind of bootstrapping approach could be achieved without relying on language similarity. The trick seems to be to tap into pre-existing adult skills or knowledge - e.g. logic, map reading. I think this is what sets LLPSI apart from “immersion” approaches like Rosetta Stone, which are based on the erroneous assumption that adults (do / should) learn language as toddlers do.