I view it as three different film series. The cast (or at least the star) makes the series. If you make enough films, there's going to be turnover behind-the-scenes, so the star is the keeper of the flame.
Waaaaaaaayyyyyy back in 1994, there was an official Star Trek: Generations magazine that came out that was put out by Starlog. I might or might not still have it (probably not), but I remember one of the actors (I think it was Marina Sirtis) saying "This isn't Star Trek VII. It's really Star Trek: The Next Generation I." I think that makes sense.
Calling Generations "Film 7" is like calling an episode of one of the new series "Episode 750". Generations is the seventh film of The Franchise, but it's the first of its particular series. It's common to number the films the same way we number TV series. In 2002, we had the fifth series (ENT) and the 10th film (NEM).
But I think films are more like "seasons". In the '80s and '90s, for instance, Star Trek had a string of long-runs. Six films with the TOS movies, and TNG/DS9/VOY all had seven seasons each. If TOS had seven films centered around their cast, it would look more obvious. Because then it would have the same number of films as the '90s shows had seasons.