Your logic is correct. But the operation of splaying is only done during insertion and searching and not during traversal. When you insert or search a node it is moved to the top(made as root node) so that it would be accessed quickly thereafter.
This is impossible if priorities can be compared only. The problem is that a constant-time merge can be used to simulate insert in constant time, which, since delete-min also is constant-time, violates the known lower bound on sorting.