An unshootable bow is just a piece of kindling, old lemon wood bows are pretty common, little if anything special about them.
Get that sucker shooting again. If it is long cut it off a couple of inches. If you want to back it, flatten the back and add a 1/8" strip of hickory glued in a reflex then retiller.
There are some special old bows that are worth keeping intact even if they are wall hangers, others will end up in a yard sale for a couple of bucks after you are gone, which in my opinion is about what they are worth.
I have an old yew bow that actually had a piece of electric fence wire for a string when my friend bought it at an auction for $2. No sapwood, very narrow, broken on one tip and twisted limbs to boot. Some day I will bring it back to life with a strip of bamboo and little TLC.