I'm certainly not an Olympic shooter, and not even a true target shooter. I shoot at targets for fun and to practice for hunting. I am a fairly good shot, but there are many thousands of archers who shoot much more accurately than I do. I have somewhere between 40 and 50 trad bows and have shot a lot more over the last half century. Most of my bows have wood (including bamboo) cores, but a number of my newer ones have foam cores, and some of those also have carbon backing. The "feel" of a bow encompasses many things. The most important factor is good design. However, materials do make a difference. Foam cores and carbon backing cannot make a poorly designed bow shoot well, and some extremely well designed bows are not amenable to the use of foam and/or carbon.
One example of a bow with an outstanding design that could not accommodate foam cores (or carbon backing at my draw weight) is the new Brackenbury Peerless static tip recurve. The Peerless shoots at least as well as any bow I have ever had the privilege of shooting. Bill Howland experimented with the use of foam cores in the bow and was unable to adapt the foam to the design. I would put the Peerless up against any other bow made as far as the overall balance of performance, smoothness, stability, noise level, speed, and any other characteristics that archers find important in a great bow. My Peerless has wood (I'm pretty sure the wood is bamboo) cores which obviously have been used a very long time in the construction of bow limbs. Foam cores would degrade the performance and other great characteristics of the Peerless due to its design, and in my draw weight, so would carbon backing.
I'm sure there are other bows like the Peerless that are not conducive to the use of foam and/or carbon, but for those that are, these materials can definitely enhance the feel and to some extent the performance of the bow. There is no doubt in Bob Morrison's mind, or mine, that his foam core bows feel better and perform a little better. With carbon backing, they perform significantly better and feel great. As to whether the limbs will degrade quicker over time than wood cores, I can't really say, because I have not had foam core bows for more than a few years. I can say that I prefer my Shrew Classic Hunter longbows and any of my Morrisons (either longbow or recurve) to have foam cores and carbon backing, and I say that with having more boo core bows in these bows than the foam or foam/carbon models. I also disagree with those who think that a draw/force curve can adequately describe the smoothness of a bow, either in its loading (draw) or its unloading (release). The same thing goes for fly rods, since that analogy has been used in this thread. There is a good reason why bamboo rods are primarily collector's items, while carbon fiber rods rule the fishing lakes and streams.
Both natural wood core bows and foam core bows are great. Just pick the one you like and enjoy it. I expect that either material, if used by a good bowyer in a well designed bow, will last any of us a lifetime if we properly care for the bow. And depending on the bow and its design, either one of the materials may be a better choice than the other. If in doubt, ask the bowyer which material he thinks will work better in his bows.
Allan