I have not read through every single post, honestly I got tired after reading the first page with only advice on how to shuffle and cut.
Now on to your question from the OP -
I had to deal with this exact same situation, the only difference I had was the individual was BFFs with anyone, but someone who was friends with a few & played in our league for several years.
I approached him honestly and directly. Told him others suspected him of cheating, came to me and I observed his actions.
This is pretty much exactly how it went down -
Me: Other players have come to me feeling uncomfortable that you are cheating. So much so that one player recorded on his phone your actions with the deck. Shuffling/Collecting Cards/Cutting/etc etc
After reviewing his video and then paying attention to your actions myself I feel there is enough of something going on that I am not comfortable with you continuing to play in our group. Although I cannot say 100% you are cheating, something isn't right and it is my job to protect the integrity of our group and I will feel more comfortable if you are no longer involved.
Him: I am not cheating, other players are just jealous because I am better than they are. They are always trying to blame the player who wins the most blah blah blah.
Me; I understand you feeling this way and all of that may be true, but from what I have observed I feel it is in the best interest of my game that you do not attend any longer.
After this happened I told all the players the same thing. I believed he was cheating but did not accuse him or tell anyone he was. I told them the same thing I told him. I observed, felt uncomfortable and to protect my game and all the players in it I felt it was my duty to ask him not to return.
There were players who were friends with him, some who still are but I think most felt grateful that I was looking out for the best interest of the game and for them as players.
I did not lose a single player over it, in fact I know I would have lost a few players if I did nothing.