If you use persistent data-structures the way they are meant to be used, they can and will be faster than a simple array.
Both have valid use-cases, but they are often distinct.
Even the one you are making is... tough. Persistent data-structures are not a sure fire win against mutable versions. Again I find myself refering to the DLX algorithm. The entire point of which is that it is easier and faster to permute through different configurations of a data structure then it is to have all instances instantiated in a persistent fashion.
Does this mean that persistent data structures are without use? Of course not! Again, you may not be optimizing speed of execution. Which is perfectly valid!!