Why doesn't this schmitt PWM circuit work as expected?
I'm afraid your simulator is lying to you, by failing to take into account the actual input characteristics of the 74LS14.

According to the datasheet (Section 6.5 on page 5), note the values for IIH (current, input high) and IIL (current, input low). In particular, note the huge discrepancy between the two values, and that IIL is on the order of half a milliamp! There's no way you're going to be able to get that much current through your resistive divider.

I think the most direct way to address this would be to substitute the 74HC14 (or 74HCT14) for your 74LS14. These CMOS chips have much higher input impedance values, and the behavior of the circuit will more closely match your simulation.