Meaning of discontinuity in a diode circuit, analyzed using a simplified model for the diode
Yes you are relying too much on the simple model. 

Since Ipk is 1mA but then drops to 0 after 5ms,  the cap with no load R, is able to charge up to about 2.55 V from 3V . This means the diode voltage may have started near 0.6V from the 1N4148 typ. datasheet but reduced to 3-2.55= 0.45 V after the 1st sine peak.  

This corresponds to a forward current of approx 50uA which is approx, what you simulated. 

Then on the 2nd cycle another 40uA peak is conducted and the voltage rises another xx mV  and on the 3rd peak it reaches ~2.6V  or a diode Vf =0.4V 

For this test the knee voltage Vk  appears to reduce with current but we know it also rises with current. So the thing to remember is what is the Vk &amp; tolerance  @ some specified forward current. I use 0.6V@ 1mA @ 25’C 

Get used to reading datasheets  for the VI curve and keep in mind, the tolerance @25’C is due to  a series bulk resistance  due to process controls and Vk (@ x mA) is fixed, but can shift negatively with rising temp (it’s also a thermometer) 

For low currents it has a  log VI straight line  from Shockley’s Equation then a linear straight line  scale from bulk series resistance near rated current.