Welcome, Kathleen!
Many Easy Entry carts have metal shafts. I am not a fan of them because they will twist around the horse before they break. I don't prefer many hollow pipe shafts either. They are cheap and bend easily. That being said, your better brands of carts with metal shafts will run at least twice what other carts cost. Someone else can chime in on brands. We only have wooden shafts/carts other than our marathon vehicle.
Wooden shafts should NEVER be oak. Whoever told you that is off their rocker! Oak is strong, but when broke it splinters and risks poking that into the horse! Wooden shafts made by a knowledgable builder are almost always hickory! When hickory breaks, it breaks "clean" almost like someone sawed it off. Here is a photo of a broken shaft. The owner cut away the leather or vinyl shaft trim, but the break is "untouched".
All other parts close to the horse are also made of hickory: singletree, circle bar, cross bar, and wheels. That is why you may see some wooden vehicles with different grains or different finishes because of the different woods taking the stain differently. I personally like wooden shafts BECAUSE they break. I don't want my horse in a twisted up mess. Now, Combined Driving Event vehicles are a whole other subject. See above.
It's good that you are thinking about this ahead of time. The more you drive, the more it is not a matter of "if" something will happen, but "when".
If you are prepared, the "when" will be less dramatic!
Myrna