I would take a sharp stick over a AR-10. At least the stick is reliable....
FAL vs. M1A - Being fortunate enough to own both (in duplicate), I would definitely pick the FAL for a SHTF situation. It's just a better all around weapon IMO. Don't get me wrong, I love both of my M1As but the FAL just has a lot of advantages (ergonomics, function, parts availiblity, price, easier to clean, easily adjustable gas system to handle various ammo, etc.) Although a SAGE/M1A setup is pretty nice...but I can buy a complete FAL for the cost of the SAGE stock alone (without a M1A to attach it to)
If I had to buy one of the two now, I would build up a fixed 10-round FAL (now that kits are widely available once again). You can get that Tapco Stripper Clip Scope Mount along with a short scope (like a Aimpoint, EOTech, Elcan, MiniACOG, etc.) and you will be able to easily reload the gun without any sort of hassles...with some practice, you can probably load it as fast as sticking in a fresh mag (granted, it's only 10 rounds vs. 20 rounds...) You can get 10 round .308/7.62 stripper clips from DSA for $1/ea. Cheaper than FAL or M1A mags even before the '94 Klinton ban!