Why does non-inverting input of non-inverting amplifier need a path for DC to ground
As a general rule, all inputs of any kind need a DC path to ground. If you don't have a DC path to ground, the DC offset of the input can float to any voltage. You have capacitive coupling to the AC input, but you also have (weak) capacitive coupling to the rest of the circuit, the power lines, your body, etc. All of those can influence the DC component of the input voltage. A resistive path to ground (or a DC-coupled input signal) overpowers these other effects and forces a well-defined DC offset.

Why do you need a well-defined DC offset? If you don't have one, your input voltage could easily drift beyond the supply voltage range, saturating the op amp.