Some interesting discussions on this thread.

Can I just say that the integrating multivibrator, which the OP is using, is one of the most reliable and well behaved multivibrators going. It is ubiquitous. It uses the same principal as many successful timers, including the 555. Unlike differentiating multivibrators, which can get into a third state (both transistors conducting), like the classic two-transistor collector/base coupled multivibrator, the integrating multivibrator does not have this problem. Also, in contrast to the differentiating multivibrator, it has no chance of exceeding the voltage rating of its active elements.
The frequency of oscillation is controlled by the time of charging the capacitor up to and down, via the feedback resistor, to the input Schmidt high and low threshold voltages respectively- nothing to do with fast edges. If you put a scope on the input you will see an approximately symmetrical saw tooth waveform. The formula for the frequency, and associated k factor graph, is given on page twelve of this datasheet:
https://assets.nexperia.com/documents/data-sheet/74HC_HCT14.pdfA 100nF X7R ceramic capacitor needs to be connected directly across the supply pins of the 74HC14 chip to prevent it from oscillating.
An aluminum electrolytic timing capacitor is not advisable. A tantalum electrolytic or solid capacitor is far better. A polypropylene capacitor would be the Rolls Royce choice.
About the failure mode of the OP's 74HC14 chip, I hate to say this, but far from being uncommon, it is very common. I think the reason is that the individual gates rely on common circuitry for bias etc. The gates are also fabricated on a common substrate. Therefor, when one gate fails it takes some of the others with it. I have seen this many times and not just with the 74HC14. The other thing is that the 74HC14 was often used to interface to a bus or other difficult load, which gave them a hard time. Another cause of multiple gate failure is uncontrolled parasitic oscillation, where the chip overheats and destroys itself due to the high frequency. This can happen when the inputs and outputs are left floating.