I know a little bit about composites. Your idea has potential but not with hide glue. Now, I am being serious kid, your thinking along the right track and I am positive you can make a good product of this, but its not anywhere NEAR as easy as you think.
First off, choose a better glue. Folks dont expect arrows to last forever, expect them to break, but never expect them to fail due to materials. Resins are the best man made choice. I can not think of a natural glue that will work here.
Second, you will need your fibers to run lengthwise the entire arrow shaft. Any fibers that do not will cause breakage. The fibers MUST follow the lines of the direction of stress.
You must have consistency. Your wall thickness must be exact all the way down the shaft. Any spot to thin will cause breakage or irregular paradox and flight. Your fibers must also be tightly lined up against each other, with the glue being only a bonder, not a void filler.
You will need a way to press the fiber resin matrix firmly while it cures. This is often done with a vacuum bag.
You will need to do all this work before your resin sets up and flashes. You can slow your resin down and give yourself time by using a slow cure mix, and by doing it in the cold. You can also make a pre preg. This is done by soaking the fibers in epoxy, then freezing them until you are ready to use them. Pre preg at -70 degrees is usually good for several months.
The best process I can think of to do this is lay your fibers out perfectly straight on a resin soaked wax paper flat mat. Have them aligned so they are touching, but not piling on top each other. Then freeze these sheets after the fibers have become completely impregnated with the resin.
Remove the now frozen prepreg sheets and stack them to a desired thickness, and press them together with a flat board and heavy weight. This squeezes out the excessive resins. You need to make certain your fibers never move during this process. After the flash off has started, and the prepreg sheets are still very tacky, you will need to roll them onto your dowel form, and use a vacuum bag to apply the needed pressure to the fiber matrix to keep them EXACTLY in shape and pressed together enough to ensure the fibers bond to each other and not fibers to a glue filled void.
Some experimentation may be needed to find appropriate wall thickness to get the spine you want. I would say use other materials too, like the fibers from very stringy plants. Celery stalks, asparagus would be a perfect one, carrots, any stringy vegetable.
Anyways, its a long post, but, having worked composites in the aviation industry, I know a little bit about them. The process is simple. The hard part is making your prepreg matrix. Good luck and post what you do regardless of what it is!
