I am not the expert of this field actually, thus, I haven’t followed too much about the approaches before deep learning, but I try my best to say:

1. Using overlapping small patch for training is a conventional practice to train a machine learning model, or least square to find the weights for interpolation.
2. Now, it is changing. e.g.: In VDSR, direct quote: “Input patch size is now equal to the size of the receptive field and images are divided into sub-images with no overlap.” A much larger input patch is used.