In his article, "The Future of Back to the Future," Matthew Holker explains that Bob Gale and Robert Zemeckis have clearly stated that a sequel, remake or reboot of the popular franchise will not happen, but that the popularity of other movies that have undergone movie metamorphoses makes it "inevitable" that the same fate is destined to happen to the Back to the Future trilogy. I don't doubt that there will be a time in the future where a new incarnation of Back to the Future will come to fruition. But, I feel there is an issue with using his tie-in argument, "Where are all the hover cars? Something went wrong, and we had better figure it out before it's too late" as a starting point for a new second trilogy.
2015 - Then and Now
In Back to the Future: Part III, Jennifer Parker points out to Doc Brown that she brought back a note (the "YOUR FIRED!" fax) from the future and now it's erased. Doc says, "Of course it's erased." When Jennifer asks what that means, Doc says, "It means your future hasn't been written yet. No one's has. Your future is whatever you make it. So make it a good one, both of you."
What this means is that Doc understood that whatever future they all saw can and most likely will change. Therefore, in the real 2015, there would be little incentive for Doc, Marty, or another new character cast in a second trilogy to go back in time and try to figure out why there are no hover cars, hoverboards, self-drying jackets, etc. As a matter of fact, there are now similar things like flying cars (albeit, not hovering, gravity-defying cars), hoverboards (Hendo has developed one that doesn't work exactly like the ones in Parts II and III but, nonetheless, they are in the works). With these inventions, while not exactly the same as in the 2015 that Doc and Marty visited, it would seem that variations did somehow occur. This would only make Doc and Marty feel unconcerned that their known versions of these inventions did not manifest themselves.
Bob Gale further explains in the Official BTTF FAQ on Wikidot.com that "time travel into the future takes you to the most likely future of the moment you left." So even if you go on the theory that somehow "something went wrong" to cause hover cars and other items not to exist in the real 2015 (or in 2017, which Holker uses as the date in his article as a possible starting point for the making of a second trilogy), Gale's reasoning postulates that the existence of certain items in the future, as time passes in real time, would likely be different, yet similar. In other words, the space-time continuum extrapolates time travel to the future as best it can, based on past events, while living life through the natural course of time, without the use of a time machine, allows for a more realistic future to unfold.
Making the Argument for Another Trilogy
Now, if we exclude the plot lines from "Back to the Future: The Game," you could say that Doc's irresponsibility in living in the late-1800s without any care in the world might change Marty and Jennifer's future. We know in Part III that Doc hypothesizes to Marty, "We know that this photograph represents what will happen if the events of today continue to run their course into tomorrow," referring to Doc's name being expunged from the tombstone due to recent actions by both Doc and Marty in their run-in with Buford Tannen and his gang in the saloon. Therefore, there is a chance that if Doc continues to run around in the past doing this or that after the events in Part III, he will have in all likelihood altered Marty's present. Over the course of a century, even if Doc is careful to keep a low profile so as not to upset the space-time continuum, his altering of the past, even in small ways, could create a ripple effect that would be very noticeable to Marty. The world would instantly change around him, or worse, he would be erased from existence.
As Doc stated in Part III, "you can't keep a good scientist down." The urge to invent and create is still alive and well in Doc. His invention of the refrigerator in his blacksmith shop is an example of this. There is just no way he and Clara are going to live a mundane life with two kids in the late-1800s/early-1900s. History will change.
That's where a new starting point could begin at the end of Part III. After Doc leaves Marty and Jennifer behind in 1985, they are suddenly "erased from existence" and Doc is left gallivanting in the Old West with his newly established family, unaware of the new future he is destroying/altering/creating as a result of living in the past. A clean slate is established...one which opens the door to an entirely new trilogy that could still bring in old characters (Doc included) and create new ones that still tie in with the original trilogy.
Of course, there will be arguments to the effect that if Doc alters the timeline in the past, he should cease to exist just as Marty did. But we're talking fiction here. Maybe the ripple effect has characteristics that were as yet unknown to Doc, where the ripple of time skips over Doc's existence. That knowledge later becomes known to a descendant of the Brown Family -- a young inventor who becomes fascinated with his ancestor's "fictional" books about science and time travel, likening his fascination to Doc's keen interest in the writings of Jules Verne.
A New Beginning
Around 1987, a young aspiring inventor in his late 20s -- the great, great grandchild of Doc Brown -- finds a box of Doc's old writings in his attic and discovers material about how to build a time machine. At first, the inventor thinks he's discovered unpublished fiction that Doc never used in his writings. But he soon figures out that the writings are more than fiction. Over the course of 30-plus years, he uses the writings to create, you guessed it...a working DeLorean time machine. DeLoreans still exist in the 21st Century. But Mr. Fusion doesn't exist, because the real 21st Century is very different from the one that Doc and Marty visited in 2015. So the inventor comes up with some other form of power to create the "1.21 jiggawatts" of electricity he needs to travel through time in his newly invented time machine. Enter a descendant of the McFly Family, albeit not Marty McFly, Jr., but some other direct descendant, that befriends the old, eccentric inventor and...well you get where this is going.
To Be Continued (?)
This would seem to be a much cleaner way to start a second trilogy. Of course, fans of the original trilogy have said, and will continue to say, that without Doc and Marty, it's not Back to the Future. Honestly, a part of me also feels that way. But if it's inevitable that Doc and Marty's legacy should continue into the future, that continuation will most likely be in a form that creates new characters as descendants from the original characters, maybe using some of the other characters from the original trilogy in cameo form, and, sadly, leaves some of the beloved characters from the original trilogy behind.