If we ignore the capacitance of the input pin, then it would instantly follow the voltage of the send pin. A voltage drop over the resistor only happens if a current is flowing though it, which cannot happen without any capacitance the receive side. Without current a resistor is just a wire.
A floating pin doesn't directly have its state randomly changes between LOW and HIGH. That can only be the result as the voltage on that pin is changes by the noise. We need the noise to not be directly in the logic voltage range, so that the state of the digital input pin is not totally dictated by noise. That often the case. Also the additional capacitance added by the sensing foil helps with filtering the noise.
The actual measurement works as follows: We set the send pin to a known state and wait a bit (to fully load or unload the capacitance on the pin). Then we change the send pin to the other state and count the time until we see that change on the receive pin. That is when the capacitance of the pin is loaded. Then we are doing that procedure many times, averaging the measured time. What the receive pin does outside of the measurements is not important. During the meausrement its state is mainly dictated by us setting the send pin. Though the library might use some optimization to get a better reading. You could read the libraries code to learn more about that.
