Wolverine sort of gives the X-Men that edge, that "nastiness", the comic book trope of the savage who has to or is forced to walk the straight line. Logan/Weapon X/Wolverine gives them an extra aura, that guy who will ask tough questions or call out another X-Men's stupidity. Wolverine doesn't suffer fools gladly because in his long, mostly patched-work memoried life, he's worked with extremely brutal, efficient, and very violent individuals like Deadpool, Mystique, Saber-tooth, so Xavier can't bullshirt or order Logan around like he can Summers. Because, IMHO, Xavier sort of suspects or believes Wolverine's past may have something to do with one of the X-Men's core missions, and that's the history of federal government knowing about the existence of mutants far earlier then they've admitted and experimentation on mutants like Wolverine.