Let my preface my answer with a quote from Wally Schirra after Gemini 6A:
"Somebody said ... when you come to within three miles (5 km), you've rendezvoused. If anybody thinks they've pulled a rendezvous off at three miles (5 km), have fun! This is when we started doing our work. I don't think rendezvous is over until you are stopped – completely stopped – with no relative motion between the two vehicles, at a range of approximately 120 feet (37 m). That's rendezvous! From there on, it's stationkeeping. That's when you can go back and play the game of driving a car or driving an airplane or pushing a skateboard – it's about that simple."
What Captain Schirra was saying was that there is a huge difference, in terms of the piloting task, between rendezvous and stationkeeping.
The actual rendezvous procedure initiates upon the liftoff of the "chaser" vehicle (in fact, this can be considered the first of what will be numerous "rendezvous burns.")
Once the rendezvous has proceeded to the point where the range between the "chaser" and "target" vehicles is down to ~1000 ft or less, the chaser vehicle's pilot will find that, if all relative motion with respect the target vehicle is reduced to zero (or, almost zero), the task of maintaining a steady position (again with respect to the target vehicle), or staionkeeping, becomes quite simple - as Captain Schirra put it, "That's when you can go back and play the game of driving a car or driving an airplane or pushing a skateboard – it's about that simple."
In other words, once the chaser vehicle has attained the conditions necessary to set up stationkeeping, the effects of orbital mechanics become almost imperceptible - and get less noticeable the closer the two vehicles are to each other...
To put things into perspective, NASA's Rendezvous Crew Training Handbook (dated November 1998) states that, for the Space Shuttle Orbiter, stationkeeping, when on the Vbar at a range of 1000 feet and in a circular orbit of 160 nautical miles, should require no more than ~70 lbs. of propellant per orbit. That's pretty low. Alternatively, said reference also states that, if said stationkeeping is instead set up at 40 feet on the Rbar (with the same target vehicle orbital parameters), said prop consumption should be on the order of 100 lb/rev.
FYI, one can generally infer that, the simpler the piloting task, the lower the rate of prop consumption.
So, to answer the question, I'm not sure how much actual decent simulator time the crew of Gemini 4 spent in rendezvous practice, but it appears as if the simulator referenced in the OP probably did a good job of simulating stationkeeping and the docking task (said docking task being essentially a lot like stationkeeping, with the added elements of setting up a low magnitude closing velocity vector pointed toward the target vehicle's docking mechanism). Note that, in the NASA paper referenced in the OP, the initial conditions for a docking simulator run were set up so that the chaser vehicle was 55 feet away from the target vehicle, with no rates (see pg. 9 of said reference) - pretty much in line with relatively simple stationkeeping situation. However, the crew of Gemini 4 never even came close to achieving a stable stationkeeping situation...
On a side note, of interest in said NASA paper are the pilot ratings commencing on pg. 23 of same...fascinating!