No you are not better off selling the guitar for parts - this is a fairly common break on
Les Pauls due to the way the grain runs at an angle through the headstock - a good bang on the back of the headstock and voila - you see the results.
You are pretty fortunate in that it is a very clean break - no splintering, no missing bits etc - a decent repairman should be able to fix that WITHOUT any cosmetic touching up for somewhere around $50 - $75 I would have thought.
Here's a couple of pointers - don't try to remove the headstock, no point, don't get talked into adding graphite or ebony or maple reinforcing dowels - no point! The only glue that should be used in a repair like this is either Fullers Titebond, or a hide glue - most repair shops will use Titebond as hide glue stinks and is very messy. Don't tinker or fiddle with it - the more grease, dirt, finger oil and so on you get in the crack, the harder the repair becomes.
DON'T use, or be talked into using any form of epoxy glues, araldite, or any other form of 2 - pack glues - they don't work well with a break like this, and you have very chance of the headstock falling off again, except the luthier will charge you considerably more to remove all the epoxy glue which didn't stick the wood in the first place. Likewise Super Glue is not an appropriate adhesive at all for a task like this. Unless you are pretty sure of what you are doing (and as you have asked, I guess the answer is thet you aren't!) - leave it to a decent repair person!
Your Les Paul is worth, what, between $1200 - $2000US? - of course it is worth repairing. Besides - you live in LA, the home of the guitar tech!