Biasing the Schmitt, or standard CMOS gate, will subject the gate to noise, and
pickup to noise unless you scale the divider to lower Z to avoid pickup. That
requirement loads oscillator unnecessarily, and burns power.
The tolerance of thresholds on CMOS Schmitt awful as well, so no guarantee
what duty cycle you will wind up with.
Best to create a schmitt of your own with reference and fast amp and precision
R's.
Regards, Dana.