Training Neural Networks (NNs) with Genetic Algorithms (GAs) is not only feasible, there are some niche areas where the performance is good enough to be used frequently. A good example of this is Neuroevolution of augmenting topologies or NEAT, which a successful approach to generating controllers in simple environments, such as games.
In the more general case though, the approach does not scale well to large, deep networks with many parameters to tune.
Genetic algorithms and other global searches for optimal parameters are robust in ways that gradient-based algorithms are not. For instance, you could train a NN with step function activations, or any other non-differentiable activation functions. They have weaknesses elsewhere. One thing relevant in the case of GAs used for NNs, is that weight parameters are interchangeable in some combinations but heavily co-dependent in other combinations. Merging two equally good neural networks with different parameters - which you would do in cross-over in a GA - will usually result in a third network with poor performance. NEAT's success is partially in finding a way to address that issue by "growing" the NN's connections and matching them up between similar neural networks.
Gradient-based approaches are much more efficient. In general, and not just in domain of NNs, if you can calculate gradient of a function with respect to parameters, then you can find optimal parameters faster than most other optimising techniques. An accurate gradient guarantees at least a small improvement from a single evaluation, and most other optimisers fall into a generate-and-retry paradigm which cannot make that kind of guarantee. The weakness of tending to find local optima has turned out not be a major hindrance for the loss functions in NNs, and has been tackled with some degree of success using extensions to basic gradient descent such as momentum, RPROP, Adam etc.
In practice on a large multi-layer network, gradient methods are likely orders of magnitude faster than GA searches such as NEAT for finding network parameters. You won't find any GA-trained CNNs that solve ImageNet, or even MNIST, where the GA has found the network weights unaided. However, GAs, or at least some variants of them, are not 100% ruled out. For instance this 2017 blog reviews recent papers including Large-Scale Evolution of Image Classifiers which explores using GAs to discover NN hyperparameters which is an important task in machine learning, and not very tractable using gradient-based methods.