Well, I can get it to oscillate in a sim with SwitcherCAD, but I don't have a model of 74HCU04. I did a best guess by using the behavioral inverter they provide.
I don't think the oscillation will be at the resonant frequency of the tank, because the phase shift at resonance is 90 deg. You would need another 90 deg. to add to the 180 deg of the inverter. I think it depends on the 470k-Cin time constant for some phase shift. The 5ns of the 74HCU04 adds a tiny bit. I think the tank will be off resonance. I got it to oscillate (simulated) at resonance by adding 50pF to GND on the input of the inverter.
One thing was baffling. If I used .IC v(out)=0, where out is the L-C junction, I got a different duty cycle from the inverter than if I omitted that initial condition.