Low precision oscillators such as silicon oscillators can usually be used for processor clocks. They can also be used in a number of situations where an AC signal is required but only the envelope is measured. For instance, IR remote controls use a 38 kHz carrier, but the exact frequency can vary be several kHz -- the modulation is only so they can do AC measurements instead of DC. The same goes for basic ultrasonic rangefinding -- only the round-trip time of the pulse is measured. Switch mode power supplies and PWM drivers are also pretty much insensitive to frequency.
For measurement applications such as capacitance sensors the answer depends on the resolution you need. A low precision oscillator is still higher precision than most other electronic components, so maybe it doesn't matter much. On the other hand, cheap and accurate crystal oscillators are easy to come by. If precision matters at all, people tend to use them and then try to make the measurement depend on as little as possible except the oscillator frequency.
The most important time hobbyists are likely to require the greater precision of a crystal oscillator is in communications protocols. Many communications protocols have very strict tolerances on the clock frequencies. For instance, you might have a requirement that the transmitter and receiver should not have their clocks drift by more than 1/4 of a bit within a single packet. If you support a 1500 byte packet and do the math, your find your clocks need to be better than 10 ppm error over all possible conditions including operating temperature, aging, and supply voltage. High speed protocols use clock recovery, where the receiver's clock is actively synchronized to the transmitter by measuring the timing of the signal transitions, but even then you want to have the clocks start as close together as possible. Communications applications are also
Timekeeping requires an unintuitively (to me) high precision clock. For instance, that LTC6930 has a temperature coefficient of .001%/C. If you make a watch out of that it would lose (or gain) over 6 minutes a month simply due to the temperature shift between your wrist and your workbench.