Charles Xavier occupies a unique position within the Marvel Universe. As the founder and moral leader of the X-Men, Xavier's dream for a world where mutants and humans could co-exist has been long considered a foundational aspect of the X-Men's role as heroes. Progress has been made in bursts -- juxtaposed against attacks on the community.
Eventually, however, the mistakes made by Xavier and his peers came back to haunt the next generations of mutants, leading some of his most prominent former students to rise to the occasion and become figureheads themselves. In the modern-day, Xavier is a contentious leader in Krakoa as a member of the Quiet Council but is notably no longer in charge of the X-Men or even really his legacy. But which mutant has truly stepped up and set themselves up as the heir to his role in the X-Men universe?
Figuring out his true heir is a tricky question, considering the importance of some of the competitors for that title share. Cyclops and Beast have both in the past made good claims to being Xavier's heir, with Cyclops -- the first official X-Man and one of the most enduring leaders of the team -- being an obvious choice. Other realities have even codified this, such as when the Cyclops of Earth-X began going by Professor S in-direct reference to his deceased mentor. But the 21st century has seen Cyclops embrace his more pragmatic side to contend with the threats that arose to derail the mutant population's growth. This eventually resulted in him becoming more militant, even being regarded as something of a villain following his brief corruption into the Dark Phoenix. He now serves once more as the leader of the X-Men.
Notably, Cyclops has turned down positions of leadership on the Quiet Council since then, instead preferring to serve as a defender of Krakoa. This positioned Beast into a more overtly important role as Xavier's heir. Beast remained a more peaceful figure in the mutant community when Cyclops and others increasingly broke away from humanity in the fallout of Decimation. Beast's commitment to education was a factor in this direction, and his connection to the Avengers has been a major element of the character for years. But like Xavier, Beast increasingly puts his own decisions before others (such as dragging the time-displaced O5 X-Men into the present and threatening the time-stream). He's even embraced his darker side, becoming Krakoa's ruthless tactical leader of X-Force in contrast to his former pacifistic side.
These are the sort of directions that would have concerned the classic version of Xavier. Neither of them has truly become the figurehead for working with humans and mutants coming together, an ideology that's largely fallen to the wayside in the Krakoa era. Even Cyclop's X-Men are designed as superheroes above all else, with positive effects happening to the world more as a happy result of their primary missions. While Warpath might be committed to keeping the dream of co-existence alive, the best candidate for being the next Xavier might actually be Wolverine. A brutal loner when he joined the X-Men, Logan's appreciation for teaching and being a mentor has become a major factor of the character.
He even served as the head of a mutant school modeled after the Xavier Institute, helming the Jean Grey Institute for a time. His experience with other human heroes makes him a bridge to the greater world, and his growth as a character has seen him genuinely gain friends and family instead of just operating behind the shadows. Wolverine works with Beast as a member of X-Force, but has committed himself to this dark path almost solely because it allows him to clean up the mistakes of others. Logan is one of the best examples of how ways of the X-Men can turn a killer into a teacher -- which has become a major undercurrent of the franchise, especially lately.
Wolverine understands, better than almost any other hero, how important the kind of work Xavier used to do really is -- and has been shown in multiple possible futures to embrace that path himself going forward. While he may still be Krakoa's weapon when they need him, Logan has grown to perhaps the best possible replacement for Xavier in the Marvel Universe as a figurehead and leading educator of the mutant population.