My own personal take on TAS is that it's quite faithful to TOS. Possibly uniquely so.
If we look at other 'live action' shows that got Saturday morning animated adaptations, we can see just how badly most of them fare. Usually in the ways of adding "kiddie fodder" to the existing format that do nothing but take away a lot of the charm of the originals. Introducing a cute comical sidekick, for example, or perhaps by blatantly disregarding the original premise in favor of introducing something completely wacky and off-the-wall that would never have been seen in any live-action version, but which was deemed okay because "Hey, it's just for kids, they won't care!"

This kind of thing was like THE standard for cartoons of that vintage, and shows based on live-action versions were no exception to the rule.
Now look at the extent to which TAS perfectly encapsulates Star Trek, which to my mind far outweighs the few occasions when it
sometimes dropped it's guard and lost the plot. To this day, I remain amazed at the detail to which Filmation reproduced the Enterprise interiors and exteriors, and remained faithful to the original designs and asthetics of the sixties show. Also, we can not doubt that the presence of so many TOS stalwarts behind the scenes was a factor in this as well. Most cartoon shows based on live-action properties never had the benefit of the original live-action show runners behind the scenes, but TAS had Roddenberry and Fontana on full-time creative duties, while the likes of Gerrold, Kandel and Peeples provided scripts. These guys were Star Trek veterans, and it is down to them that TAS feels
so much like an extension of TOS, rather than simply an animated adaptation of it. And that, my friends, was truly remarkable in the annals of 1970s/1980s animated adaptations. I can't think of another show like it.
