"Brute force" would be trying random weights and keeping the best performing model. Backpropagation is compute-intensive but I wouldn't call it "brute force".
What? Either option requires sufficient data. Brute force implies iterating over all combinations until you find the best weights. Back-prop is an optimization technique.
No, a large dataset does not make something brute force. Rather than backprop, an example of brute force might be taking a single input output pair then systematically sampling the model parameter space to search for a sufficiently close match.
The sampling stage of Evolution Strategies at least bears a resemblance but even that is still a strategic gradient descent algorithm. Meanwhile backprop is about as far from brute force as you can get.