Your argument is that gravitational waves must be quantised because otherwise arbitrarily small amounts of energy could be transferred to a gravitational wave and therefore all objects would be steadily losing energy to gravitational waves.
Your argument doesn't work because (in principle at least) all objects with an oscillating quadrupole in their mass distribution are indeed steadily losing energy to gravitational waves. However the rate of energy loss is too low for us to measure.
When we say gravitational waves are quantised we mean that energy can only be transferred to or from them in discrete chunks that we call gravitons. However we expect that the energy of an individual graviton is so small that it is likely to be forever beyond experimental measurement. So any system we can observe is operating in the classical limit and the quantisation of the gravitational wave will have no observable effect.
Objects cannot just lose energy to a gravitational wave because gravitational waves are generated by an oscillating quadrupole in the mass distribution. The simplest example of this is a rotating dumbbell, which is basically what a binary star is, and indeed we have measured gravitational wave emission in the Hulse-Taylor binary pulsar system.
But the coupling of the gravitational field to mass is extremely weak so unless you have extreme sytems like a binary pulsar the amount of energy lost to gravitational waves is undetectable. For more on this see Is it possible to produce gravitational waves artificially?.