Does increasing kernel size in a CNN result in higher accuracy on the training set?
I'd say there is no direct relation between the kernel size and the accuracy.

If you start using larger kernel you may start loosing details in some smaller features (where 3x3 would detect them better) and in other cases, where your dataset has larger features the 5x5 may start detect features that 3x3 misses.

So, I'd say "no".

Anyway, if you add a second convolutional layer on top of your first you'd start having something that's close to larger features. E.g. your small kernels in the first layer would detect small features but the second layer would detect features which are composed of several features from the previous layer.