I know that capacitor slows the rise of voltage, but why is this not happening in the LTspice simulation?
I got the input voltage at the output instantly.
Go to your simulation setting and check the "Start external DC supply voltages at 0V".
As noted in the comments below, there are several ways to solve this. Initial point solution and detect steady state are both methods to speed up a simulation at the cost of dynamic behavior at startup. I personally don't like options like these and "cheat" with a spice directive like ".ic V(Vcc)=0" if it's charged externally or use a voltage source which does its own ramp.
Be aware of the shenanigans going on behind the scene in LTspice and find a method you are comfortable with.