I'd say the only problem with a bow/crossbow is eventually you're gonna be whittling sticks to shoot out of it. Arrows, bolts, and broadheads are kinda pricey, how many can you afford to have on hand?
I've actually got arrows that has lasted for three years and going, at one point and time I was shooting a lot of 3d club shoots and a lot of target practice, as long as I was carful and stuff they would last a pretty long time. But you are right there would come a point in time that would be the case but I think it would last longer than any ammo or chemicals for a PB.
I don't know I think it will all boil down to being prepared with not only the right equipment but also the right knowledge. This is one reason why I started doing my own processing, and am getting ready to do my own smoking and other long term food storage technique, to Learn so I know how.
Unfortunately we have allowed far too many of these valuable once common skills fall by the wayside.
I don't actually think we will find ourselves in knee deep in zombies

but it isn't going to be pretty. It will be those that can and will be able to take care of themselves and or has a valuable skill to offer others in trade that will come out alive or at least the best chance to do so. A good flexible mind will be the most valuable tool in any long term survival situation no matter the cause,
If it does go sideways then I will hole up and let the idiots kill themselves out while me n mine and those like minded stay together and share our resources and build our walls . And since I know someone will ask,.,. When what's left of those idiots come a calling what then? You would be amazed what you can do with a few household chemicals and sugar

let em come
I've made my own knives and blacksmithed some of my own tools, can build my own furniture and work a wood lathe. Successfully set and run traps and have some wild plant knowledge, know how to grow and can my own veggies and make my own shine.......(for medical purposes of course

)
Sorry this kind of got off it's topic,
You guys talking about 22 cal Springer's and stuff try living a week on rodents and birds before you decide to stick to your guns (pun intended). I will give you a valuable tip look up any qualified survival teacher and ask if it can be done. The body needs fats and other nutrients that they don't have to survive. Short term yes but long term no way