There are multiple ways to skin a cat, though (perhaps literally; this would be the post-apocalypse, after all!), and you're right that there are numerous ways to put transistors to use besides building full-blown CPUs. One of the key advantages of a general-purpose CPU is that it's general purpose and can be made to do all sorts of different things, but there are certainly plenty of cases where that ain't necessary and you'd need a fraction of that capability at most.
Still, they'll probably go hand-in-hand. "Chips" are just discrete components, whether crafted from a single chunk of silicon or itself built out of discrete components and treated as a single discrete unit. Building a whole general-purpose CPU from individual transistors is much easier when those transistors are already arranged on a little board you can plug into your bigger board. Chances are that no matter if someone's building a whole CPU or something more special-purpose and limited, that someone will be doing so in terms of already-assembled-and-composable gates rather than transistors directly, if only for the sake of one's own sanity.