I also use a small, fine mesh skimmer. Or a ladle. Or a large cooking spoon. Or a saucing spoon. Or a regular tablespoon. Or nothing, doing it the way Pete does.

In other words, it's an easy operation. Nothing to get your knickers in a twist about.
In the stewarding department at school, we made c.b. from the partial pounds returned from classes: dumped them in a pot, melted and allowed to boil, skimmed if someone was nearby to do it (not usually, though), ladled off the clear fat and left the milky stuff and bits of wrapper

in the bottom of the pot. And a place where I used to work made huge batches of a blend of butter and olive oil. If someone had the chance to skim, we would. If not, no problem; we just let all the solids sink eventually and then ladled off the clear fat once the boiling stopped. That made more work for the dishwasher

, since the solids did cook onto the bottom of the pot. But the resulting product worked -- and kept -- just fine.
I am about to make a new batch of c.b. -- just finishing the 2 pounds/1 kilo I made in March of last year and kept in jars in the fridge. Whenever I need some, I just dig in with a CLEAN knife or spoon. The stuff keeps fine, since you have removed the elements that are likely to go bad.
I also save the stuff I skim off, to put on vegetables. However, I don't save the milky part, or the cooked solids.
Finally: I just read somewhere -- might have been Shirley Corriher (?) talking about advice she received from Paul Prudhomme (?) re: clarifying butter. The advice was: do NOT put the whole thing -- clear fat and milky stuff still on the bottom -- in the fridge uncovered. The fat can pick up odors that way. Make the little extra effort to ladle off the fat while it's still liquid.