Integration vs component testing has almost nothing to do with functional vs state/IO heavy work. Instead,it has everything to do with the amount of effort you spend in specifying your components and writing test cases. If we're building a feature where componentA must call componentB with some data structure to achieve some goal, we can formally specify all valid inputs of componentB and their semantics, and write tests for each combination etc; and then have componentB religiously stick to the same; or we can agree in more informal terms on the expected input values and rely on integration testing to put them together and make sure we are achieving the right result for the expected inputs to componentA.

For some problems, the formal specification is tractable and even necessary. But for many complex problems, it is either not tractable (the input is too complex, you would need on the order of magnitude of componentB to actually specify the semantics) or its just not worth it (componentB is only called by componentA).

I also want to note that I'm not talking about regular types when I say 'formally specifying the valid inputs and their semantics', though I'm sure dependent types could in principle achieve this. I'm talking about cases like components which comunicate though script-like objects or configuration templates etc.