With all this talk of Les Pauls being so crappy for tuning and easily prone to breakage you have to wonder why so many players have toured with them year after year after year (decade after decade?) all over the world.
Keep a close eye on your LP and you'll never have to worry about breaking a headstock. Tons of options for easily improving tuning.
Amazing how so many LP threads end up focusing on broken headstocks when it's such a rare occurrence - unless you're a dumbass and don't know how to take care of your stuff.
Got my first black with gold hardware LP custom in 75-76. Probably owned a half dozen of them since - each for long stretches. Never broke a headstock and never suffered tuning issues for for any longer than it takes to add a little graphite to the nut slot.