And the reason why you cannot check that the program performs no I/O is because you cannot check if the I/O instructions are behind an infinite loop or an impossible condition or not.

It is definitely easy to verify that a program has no references to I/O instructions. This will reject programs like «if(false) then write("Hello")», so it is not a violation of Rice theorem (functionally equivalent programs are not treated in the same way). But you don't need to accept everything that is actually benign, you just need to make it easy to write benign code that passes the check.