Well dynamical systems aren't usually random for a start.

I suppose you can make it fit the definition for the random dynamical system, though in that case the underlying dynamical system is ergodic.

I think the article is correct that the system isn't ergodic, because it's not "mixing". A given typical trajectory won't get close to every potential point. Most go to the bottom and few go to the top - basically the point of the article.

The fact that it's biased is irrelevant. The question whether it will get close to any point is quite complicated, though with it being essentially a random walk with drift the chance that will get above its initial point is probably less than 1.

This's got nothing to do with ergodic theory though, but feel free to use that word in whatever weird distorted way you like, I literally can't seem to stop you.