There is a fundamental difference between "We don't know how to solve this practically since the best solution we have would take too much time." and "It is a problem without solution. It can not be solved at all not matter how you approach it, or even how an imaginary alien specie with technology beyond our understanding would approach it".

Also, the equivalence between the different models of computation for Turing completeness (turing machine, lambda calculus, ...) isn't about complexity classes of the problems, only wether or not the model allow to solve the same set of problems. For example searching in a sorted array is O(log n) on a Von Neumann machine but O(n) on a Turing machine.