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.
Also, look at how Xavier talks to and reasons with Logan in the comic-books, TV series, and movies, he talks to him like a professional, not some young, impulsive, irrational hot-headed mutant who hasn't lived at all.