I had the same feeling as you when I watched the video again recently. It seemed like one of the ice giants would get ejected after coming too close to Jupiter. It turns out that there's a name for this: the jumping Jupiter scenario. Outside Wikipedia, it's described in Fassett & Minton (2013) (paywall!) and tangentially in Deienno & Nesvorny (2014). Without this kind of encounter, the result of the Nice Model would not be what we see today (e.g. Brasser et al. (2009)).
This is what is known to the experts as a big problem. In most- but not all - jumping Jupiter simulations, the ice giant is ejected. One proposed solution is the hypothetical 5th giant, developed by Nesvorny (2011). It involves a sort of sacrificial gas giant planet - possibly an ice giant - interacting with Jupiter and Saturn (as per the jumping Jupiter scenario) and then being ejected from the system.
Nesvorny also indicates that certain conditions must be met. For example, if the 5th planet is low-mass and forms further out than Neptune, Neptune will - as you predicted - be ejected, although some simulations threw out Uranus instead. A similar, though less dramatic, result arises when, instead of adding in a 5th planet, the mass of the disk in increased. Neptune moves way too far out, as you guessed. A low-mass disk and 5th planet of similar composition and mass to Uranus and Neptune stops these disasters.
In Nesvorny & Morbidelli (2012), it is noted that in some cases - most notably a case where Jupiter and Saturn share a 3:2 resonance and the inner ice giant (Neptune) share a 3:2 resonance, one or both of the ice giants are ejected from the system. Only one simulation didn't lead to that outcome - and that failed to reproduce the current conditions.
On the other hand, putting the ice giants in a 2:1 resonance solved the problem, or at least led to more stable orbits. The interesting thing with all of this is that the papers claim that a 3:2 resonance between Jupiter and Saturn is more likely, rather than the 2:1, as is often cited.
I can't tell what resonance the video uses, though it says it is based off of the model of Gomes, Levison, Tsiganis and Morbidelli (2005), which uses the 2:1 resonance. Morbidelli's collaboration with Nesvorny is much more recent, though, and says it is based off of newer work (besides their simulations). So it could be that the video is inaccurate (as of current models), and new fixes (3.g. the 5th giant or the 3:2/2:1 resonances) are needed.