There's some truth to it, but it's often misunderstood. An air stone doesn't directly remove CO2 from the water; it increases surface agitation, which speeds up gas exchange. If your tank has elevated CO2 levels from injection, an air stone can reduce them. In a low-tech tank like yours with Java fern, moss, and moneywort, CO2 is usually already close to atmospheric equilibrium, so the effect is generally minimal.
Many people run an air stone only at night on a timer, especially in planted tanks, since plants stop consuming CO2 after the lights go out and oxygen levels can drop. For a 3-gallon betta tank with hardy plants, either running it at night or not using one at all would be perfectly reasonable unless you're seeing signs of low oxygen or poor circulation.