Gauge symmetry is a local symmetry that dictates a lot of the structure of QCD. If your discretization breaks it (and say, only recovers it in the continuum limit) you will have very bad renormalization and fine-tuning problems. Putting the gauge on the links automatically guarantees that gauge symmetry is respected, even at the discretized level, and jibes very well with the fact that the gauge field / gauge connection describes, in some way, how to perform parallel transport. So, it's very natural for the gauge to connect the sites together in that you need to know the value of the link in order to compare quantities on two adjacent spacetime sites.

https://en.wikipedia.org/wiki/Gauge_theory https://en.wikipedia.org/wiki/Connection_form https://en.wikipedia.org/wiki/Parallel_transport