"Objective assessment of their quality" means what most people would expect it to mean; how elements like acting, direction and cinematography rate on a scale, for the purpose of giving out awards*. It's still opinionated, but the idea is that everyone's opinion for rating purposes is based on the same criteria. Like Starbreaker noted, most people seem to rate COTEOF as the best, while later series' efforts, like Best of Both Worlds, are never seen as being quite as good. Almost, but just not there.
Honestly, I think the non-fans always say COTEOF is the best because they think we'll applaud them for it in the face of their dismissal of Star Trek as a whole. "All Star Trek sucks, but that Harlan Ellison episode is the best of the worst" kind of mentality. And that is my assessment of this list. The writer isn't really rating the quality of the episodes, just putting them on the list where they think we want them to be. I'm willing to wager that the writer isn't actually a fan, nor that they've seen every episode the required minimum of at least once each.
As for GenX, it began on Jan. 1, 1965, according to those that record such data, and lasted anywhere from four to ten years. So anyone here born in that time period is a GenXer, and whether or not we're fans of any given series is up to each of us. TOS is my favorite because that's the one that introduced me to Star Trek. But I like most of the others, as well.
*We won't go into things like special effects due to the vast improvement of the technology of such over time, making state-of-the-art from the sixties look like Ed Wood's infamous 'paper plates on a string' by comparison.