Removing Gambit's connection to crime would mean ignoring what makes him unique amongst the X-Men. Other than characters like Cable and Wolverine - Gambit is X-Man with an established universe all his own. He could (and has) maintained a story WITHOUT involving any specific X-Men lore. He has his own social circles, rogues gallery, recurring characters, and motivations. Most of which exist BECAUSE of his criminal connections and personal background. You don't fall in with mad scientists, mercenaries, assassins, gangsters, spies, and government agents by way of accounting (you can use Iceman as an example of this lol).
I agree. Gambit's criminal background is essential to his character. He needs to be a hero with a chip on his soldier. He needs to feel unworthy while being one of the X-Men's most reliable BECAUSE of that reason. He needs to WANT to be an X-Man in search of becoming a better man - not just become one because he's scared, can't control his powers, has nowhere to go, or is forced to stay against his will. It's what makes him so multi-dimensional and why we hate it when writers ignore it and make him arm-candy.