I think Slider_Quinn21 raises a good point that a reboot could alienate old fans. And admittedly, when Chris Carter killed off Mr. Y, Erika Price and the Smoking Man, he freed the show from ever having to address the myth-arc again. There is no need to delve into Colonization or the Conspiracy of Men or the Spartan Virus any further. A reboot creates a clean slate, but we already have one.
So, here is my soft reboot proposal (without Gillian Anderson):
FBI Agent Tamlin Rivers (obviously played by Summer Glau) is in disgrace after a botched meth lab raid blew up in her face and killed 12 agents. She has scarring that's only partially hidden by her hair. She is reassigned to the most dead-end division in the bureau, the X-Files. She is briefed by Skinner (who is in a wheelchair after his injury).
Skinner says that the X-Files was shuttered by Kersh but then reopened under classified directive and he can't tell her too much about the past except that Rivers will be joining Agent Scully who has a wealth of knowledge and experience in paranormal investigations.
Rivers, a skeptic, heads down to the basement and is greeted by Agent William Scully (Miles Robbins). Glau protests that William is too young to be in the FBI; he's 18, the average recruit is 30. William dismisses all this with a smile and paperwork indicating he's a genuine FBI agent. He drags Rivers into their next case.
At one point in the premiere, William calls Mulder on the phone to ask for advice. We see Mulder in his house, holding a bag of breast milk by the fridge. William says he's not sure he can pull this off. Mulder assures William that he'll be great but reminds William that he's no longer immortal or telepathic and needs to not take crazy risks.
The conversation is interrupted by the sound of Scully screaming for Mulder off camera (using reused audio from Season 8's "Without" of Anderson screaming for Mulder in the desert) and a crying baby in the background (stock). Mulder apologizes and hangs up.
As Season 12 progresses, Rivers researches Scully and guesses that Scully is some sort of outside operative from deep in the Department of Defense somehow planted in the FBI with a false paper trail; this is thrown off by multiple colleagues at the FBI claiming to have known Scully for years.
Rivers has also heard of Agents Mulder and Scully, but when she tries to look up their records, their personnel files have been classified and their names and details have been redacted from the casefiles. As the season unfolds, Rivers finds William's peculiar ignorance of procedure and protocol to be frustrating and disturbing. He barely seems to consider the FBI a job. He sleeps on a sofa in the office.
At one point, Rivers discovers that William doesn't even get paid: payroll's never heard of him and he's been living off lottery ticket winnings (which are running out).
He's immature and juvenile, an obnoxious quality for Rivers who is a professional, precise and highly experienced agent. However, he is loyal and self-sacrificing and completely trustworthy despite being an unknown quantity, and William and Rivers develop a heated rapport of respectful teamwork.
After several standalones, we get a William focused episode and a revelation: William, losing control of his powers, broke into the defunct X-Files office to try to find answers. Reading the casefiles made him feel connected to Mulder (whom he considers his father even though they're brothers) and Scully, whom he considers his mother. The basement felt like home.
In a final burst of telepathic and psychokinetic energy, William rewrote the memories of everyone in the building to think of him as Agent William Scully and assigned to the X-Files (and had them fake the records). This burnt out his abilities, leaving him only human but having found a place where he belonged.
William called Mulder and Scully who were relieved to hear he was alive and encouraged him to take the X-Files Division forward. After more standalones, the Season 12 finale could have Duchovny (and maybe Anderson) guest-star to officially hand over the show to Rivers and William.
I can't see Chris Carter accepting this. Despite all the ridiculousness that Carter has written into his series and scripts -- Carter would balk at an 18-year-old boy being presented as an FBI agent.