From my very limited knowledge of the field, neuroscience research has a very long way to go before we can anything in the way of usable intelligence out of it. A hilarious paper about this is Could a Neuroscientist Understand a Microprocessor?
(answer: probably not).
A purely statistical approach is also wrong, and fails at many things already (language understanding, policy learning in complex environments). It's pretty clear that intelligence is not just about prediction. However, I would say that prediction (and statistics) is a major part of intelligence.
Many recent advances are a combination of some statistical approach and some logical, planning based approach. AlphaGo/AlphaZero is a good example of this. Learn a policy net and a Q-net (statistics) in order to make quick judgements and estimates about the board-state/current move. Then use MCTS as a powerful planning method to systematically plan many steps into the future. This is extremely similar to how humans play board games like go and chess -- have some powerful intuitive (statistical?) ability to evaluate the current position, combined with a relatively small amount of rollouts/planning which goes deep into the game tree.
The same for many reinforcement learning tasks -- no, you don't want to just have a policy network which outputs a probability distribution of what to do next time-step, because that's a very "reactive" policy that will never be able to plan far into the future. Instead, have some sort of planner (can be hard-coded OR learned) which puts together a plan, and then use the policy net (statistics) to execute the plan. This is pretty similar to how humans act. If I want to open a box, I quickly come up with a plan (grab the scissors, cut the tape, open cardboard box), and then the execution is left to the motor systems, which have been trained for a very long time on a lot of experience (statistics).
So no, you can't do it with only statistics, but you also can't do it without statistics.