Alternate way to read Resistance temperature detectors (RTDs) with Microcontroller
What you are talking about is called charge time measurement.  There are even microcontrollers with built-in hardware to assist with this.  Usually it is intended for measuring changes in small capacitances, like those of capacitive touch buttons.

While what you propose would work, it would have poor accuracy and more complexity, not less, compared to the traditional approach of using a single resistor and a A/D.  Your method uses a capacitance value and a time value as reference.  Time can be easily measured with great accuracy and resolution cheaply, and is not a problem.  Capacitance, however, is.  Even 10% capacitors cost more than "regular" types.  %1 capacitors cost a lot more, if you can find them at all.

One advantage of your system is that it can have great resolution.  This is basically dependent on how long you are willing to wait and how fast your counter can run in the mean time.  Having a counter run at a few 10s of MHz is easily achievable in many modern microcontrollers.  So if you're after small temperature changes, but don't care that much about absolute temperature, then this method isn't quite as silly as it would be otherwise.

However, a single fixed resistor so that it and the RTD form a resistor divider, then the result of that into a A/D is simpler and more accurate.  If extra high resolution was really needed for some reason, I'd use a external delta-sigma A/D.  Those are slower than A/Ds built into microcontrollers, but with much higher resolution.  The slowness doesn't matter when you're measuring temperature and the sensor time constant is still many times the 20 ms or so conversion time of a delta-sigma A/D.