First, best time and way to disbud kids is to burn horn buds at roughly 7-10 days, when you feel the nubs enlarge to emerge. You're destroying the horn cells. Plus cauterizing blood vessels.
Cutting isn't recommended as they bleed profusely and can regrow. It is not as simple as cutting the horn. They do not handle anesthesia well, either. I would never "cut" horns on a goat as a removal method. Their sinus cavity is below the horn area. If done improperly, you risk a dead goat...by bleed out or infection.
They have blood vessels in portions of horn, developing as horn grows. There are nerve cells too. On an adult horn, only the top/tip is safely trimmed. The outer layer entire length is hard growth over a quick (think fingernail) and a porous center with blood vessels and nerves. Painful if cut/ broken.
Some successfully band the horn of adults. It's a long process and can be painful for them as process finalizes.
The important thing in disbudding is not to be put off by the screaming and smell. Most people stop burning too soon because they can't stand the kids screaming and the smell of burning hair. You have to make sure to get all the horn root or you will have bad scurs. The kids recover as soon as they are released from the disbudding box and have a bottle.