Why does wire not have large capacitive reactance?
The same formula applies. Wires have low capacitance to their surroundings so they have high Xc. Capacitors have high capacitance in comparison so they have low Xc. Thus the wires are a smaller AC load (less current flows) and capacitors are a larger AC load (more current flows).
It is no different from resistance, high resistance means small load and small current, low resistance means high load and high current.