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The "problem" with the first two James Cameron-directed Terminator films is that they were satisfying. They told a complete story and left the viewer with the distinct sense that the tale had ended, though the final moments in Terminator 2 included the visual of a dark road with a note that no one knew what the future would hold.
The threat of Skynet had been pretty definitively wiped out, but the inherent violence in people — a theme touched on throughout the movie — still existed. It was all a bit hamfisted, but it worked. The two-movie series was over.
If there's money to be made in a franchise though, we know it's never "over."
There have since been three movies released in the franchise post-Terminator 2, and each one deals with the oddity of its own existence in different ways. My personal favorite is Terminator 3, where "There is no fate but what we make for ourselves" becomes "LOL but really there's fate, just buckle up the world is going to end."
Time travel movies always get messy in sequels, and the idea that both sides of the future war just keep sending machines and soldiers backwards and forwards through time becomes a bit silly, to the point that it's been mocked in fan-edited trailers.
There is no artistic or storytelling-supported need for more of these movies; any creative spark that led to their original creation has been replaced by commerce. There are more Terminator movies because the studio thinks people will see them, not because there are more interesting stories to tell. This is why so many of the films, including Genisys, lift so many quotes and scenes from the originals. Everyone is picking over Cameron's meal, trying to suck the marrow out of the bones.
We've talked about the ending of Terminator 2 that made it to theaters, but there was actually a cut version of the ending that would have made these sequels impossible. Sarah Connor is an older woman, talking about every day being a gift. We see her grandchild, and an older version of John. Skynet was no longer a threat, and everyone was as happy as possible. It was a bit bright for such a dark series, and it's no surprise that it was cut.
Still, this version of events would have put a serious cap on any potential sequels; we know that Skynet didn't survive because Sarah was there to witness a world without the computer system. We know Judgment Day didn't happen because she lives through its date. Someone could have still played the "alternate timeline" card, but this ending would have made it look even sillier than it does with the more open-ended final scenes of Terminator 2.
As it stands, the uneven and ultimately pointless Terminator Genisys is reportedly the first film of a new trilogy.
Terminator Genisys: Debut trailer