It’s about preparations. Obviously a horse CAN win at Belmont without having won at Belmont before basic rules of logic require it. But if you’re going for the TC without having run the horse at Belmont before then you probably were preparing your horse for the Derby, which is a shorter easier track. Belmont is generally considered the hardest of the 3, and last few decades show that horses trained for the Derby can’t cut it at the Belmont, especially not with the short rest the TC requires. And the time before then when the TC got won multiple times shows that horses prepped for the Belmont CAN win the Derby, and the Preakness, and come back to Belmont and win on short rest.
It’s not nearly all it’s ALL, and it’s not prior starts it’s prior WINS. All TC winning horses won at Belmont before the TC.
I don’t think it’s the tradition, it’s the prep. Owing to the length and overall track differences the Derby and the Stakes are very different races. And putting Derby prepped horses at Belmont is like throwing a 300m sprinter into a 400m with no time to train up. But owners currently are aiming for the Derby and then if they win that hoping for the other two. If owners actually WANT to win the TC they need to train their horses for Belmont, and then prove the horse was successfully trained for it by winning at Belmont before the Derby, then go to the Derby knowing that if they pull that one off (which they might not owing to field size and the randomness that causes) the horse is in good shape to complete the TC.