Your problem is for your Opamp.
As I see, your RC value is 100nF and 1MOhms is 0.1 seconds, and your plot is a nice 1/0.1s+1 plot.
So everything is OK based on your circuit. if you want to increase speed, lower C or R to get a better speed.
Your 500 ms delay is a 5 time of time constant of this circuit, which is also shows me that it's a first order ...
Considering the whole schematic I think
the filled dots (you encircled in blue) are nets that go to a test points
I think they correspond with the tiny round pads on the back side of the PCB, but I cannot confirm this by visual inspection only)
Typically, label these nets would have been labeled on a schematic.
the open dots (e.g the one labelled VUSB) are ...