I am a professional groomer (cats and dogs) and we never ask a customer to drug a pet BEFORE we've even given them a try!! (with the exception of the owner KNOWS their pet previously needed them)
My "miracle" tool is a shedding comb. You can comb out (using a picking motion) some of the toughest looking mats in cats. The shedding comb is a metal comb with alternating long and short teeth.
I've never heard of the "tunnel clip". It is probably just a local way of saying a clip that we might call something else. ASK what it is so there are no surprises! Shaving a cat down is a very kind thing to do if the mats are too tough to be combed out. Much safer than scissoring. Cat skin is so much more "cutable" than dog skin.