I finally gave LTSpice a try and tried simulating my circuit. Here's the weird part: if I use a blank "nmos" model for the transistor, I get the curves in the first picture. This is clearly wrong, because the cap current (red) is WAY too small there - doesn't looks particularly realistic. The voltage at the gate (blue) IS acting more like what you'd expect though.
If I try replacing the "nmos" model with an ACTUAL transistor from the database, I get the second plot: the current just spikes when Vgs is around 3.6v then immediately drops back down. Even if I increase the R/C combo, it doesn't smooth out that spike any further... It obviously doesn't have my exact transistor, so I chose one which comes fairly close in terms of resistance and Vds. I have no idea how to use LTSpice yet - I just clicked on stuff, so I'm probably doing something wrong still

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