Since we know that a 555 timer configured as an astable multivibrator depends on the resistance the timing capacitor sees, what stops people from using a 555 as an ohmmeter? How does the accuracy compare to using a traditional ADC to sense resistance via a resistive divider and looking at the voltage drop?
I think generally the reason that this approach wouldn't be used is first in the answer that @cksa361 gave. Capacitors tend to vary pretty wildly over the course of one production run. One capacitor may be an order of magnitude larger or smaller than one immediately next in the line. Also, high-precision capacitors tend to be more expensive than high-precision resistors.
The traditional resistor-divider method can be easily produced into one monolithic package. You can find several of these on Digikey or other providers. Another reason that the resistors would provide a better solution is that they would have a much more predictable temperature coefficient than capacitors. This would lend itself to a high-accuracy circuit.
Finally, the frequency output of the 555 timer would probably be less useful than the straight voltage output of a traditional resistor divider. High-precision ADC chips are much easier to come by than high-precision frequency-counters.
Generally, the 555 timer would work in theory, but the practical manufacturing considerations drive the design towards the one that you see in many ohmmeters.
Within limits, it is possible to get quite good accuracy, better than possible with a DAC with 10 bits of precision if you don't need speed. While it is true that extreme resistances (open circuit and short circuit) are hard to measure, that is true in general. The main trick with using the 555 is to not rely on frequency, but use the duty cycle instead to compute the ratio of R1 and R2. That way, the exact value of the capacitor doesn't matter at all. To me, the main strength is that a really large range of values can be measured. For example, when using a 100K NTC, resistance might vary between 100K and 250 ohm. Using an ADC with a straight voltage divider makes it hard to get accuracy at both the low and the high end of the scale.