TiVo, like all such products, ran proprietary software "on top" of the Linux kernel (and "underneath", typically, in the form of a closed source bootloader). As far as I know, no company bothers to apply such DRM unless they do this. So this is kind of a distinction without a difference.

The user space interface is explicitly supported to run proprietary software. It's an operating system. The kernel's license taints things that are tightly coupled with it. Even most modules are not this tightly coupled and can deal with the exported symbols.

Similarly, the loader, BIOS, etc, are not encumbered by the kernel license.

We are specifically addressing what happens when you take a Linux kernel (or any other GPL code), modify it, and bundle it with proprietary software, in such a way that neither the free nor proprietary software function without the other. I am arguing that, regardless of the technical details, this constitutes a derived work (and not, as is commonly argued, an "aggregate").

The FSF gently disagrees with me. Tellingly, they won't commit to a clear distinction between "derived" and "mere aggregate", calling it a "legal question", but suggest that "a proper criterion depends both on the mechanism of communication (exec, pipes, rpc, function calls within a shared address space, etc.) and the semantics of the communication (what kinds of information are interchanged)." I say this is a classic nerd mistake of reaching for technical solutions to social problems - the true criterion should be something more like "can the user run these components independently?".