Is it okay to initialize all the weights to the same value as long as that value is selected randomly using He initialization?
No. All weights should be initialized to different random values and should not have the same initial value. If weights are symmetrical, meaning they have the same value, it makes it almost impossible for backpropagation to converge to a good solution.
Think of it this way: if all the weights are the same, it's like having just one neuron per layer, but much slower.
The technique we use to break this symmetry is to sample weights randomly.