Pure functions are easier to reason about (there may be exceptions of course), that's why they're interesting.
This paper is not related to side-effects - it's related to "totality", meaning all programs terminate. This could also be interesting for side-effecting programs, but it's of course much harder to verify in this case.
Due to the halting problem's existance a total language can not be Turing complete, but many useful programs can be verified.
I didn't see anything in the paper that claims that pure functions should be "all the way down", and the paper is not about side effects anyway.