I've no tattoos, but I've had a ring in my nose for about a decade now. I've primarily worked in customer-facing roles, and I can't say that any member of the public ever displayed a negative reaction to it, or that it affected my performance in the role. Of course, it may be the case that people don't react verbally or visually to this sort of thing but are still offended, so I see the point some employers have. I have had comments from the public, but they're always curious inquiries rather than indignation. But honestly, I've never seen my interaction with customers suffer because of it, even to the point where I was beating sales targets more than other employees. Same goes for friends with tattoos I have who work in a variety of businesses. I will concede that a facial tattoo might be stepping over certain societal boundaries depending on where you live.
And I don't get the whole thing about suits in business. Where does it say that unless you wear a suit you can't be professional? Where does that whole tradition come from? I look bloody terrible in a suit, no matter the cost and quality of the cut, so do many other people.