This is a weird claim to make. The three OS's that they're talking about (desktop Linux, iOS and Android) all have plenty of popular applications written directly at the Posix API level.
Heck, when we were architecting our mobile-network-enabler stack that had client side and server side components (all in user-space), it was a no-brainer for us to just pick the posix API as our base and we designed and wrote a lot of native level code that "just worked" on all three platforms.
Even if there's any sort of framework layer between the application and the system, the Posix semantics are so deeply embedded in the culture of programming today that frameworks can say things like "this API opens a new file for write" without having to explain what all that means. The very language we use to architect a solution has deep, unspoken assumptions built-in, and it's more or less based on what Posix semantics allow.