There exist many formulas that encode the program you mentioned or other sieving methods, so it is unclear what you mean by "actual formula". See https://en.wikipedia.org/wiki/Formula_for_primes
I prefer to make a distinction between recurrence, identity, and formula. Otherwise the whole discussion is moot. For example, to compute the nth fibonacci, we have Binet’s formula. We don’t need to bootstrap - we just take the golden ratio and its conjugate, exponentiate, take the difference and scale by root 5. That’s a genuine formula. otoh If one says the nth fibonacci is just the sum of the previous two fibonaccis, that’s technically not a formula - you would need to bootstrap to get those previous ones, so that a recurrence - still very useful, but not a formula in the sense you can’t turn the crank and be done.
In that specific sense, I agree with Wilf and Dudley that the procedures outlined by Mills, Wright, Wilson etc are essentially worthless as formulas. They are nice identities and recurrences, but not a genuine formula in that you can’t get the nth prime by turning the crank - you have to bootstrap with so many extraneous params it makes the whole exercise futile.
Definition of formula does not preclude recurrence relations.
Evaluation of a formula will always require some form of "turning the crank" whether it involves recurrence or not.
The evaluation of the formula lives in sort of a separate universe then the formula itself. For example the integral or the limit of something represents something as a formula, but sometimes algorithmic evaluation of such things are not possible. A formula can exist EVEN when a algorithm or "turning the crank" evaluation doesn't exist. It's a seperate thing and does not influence the definition of "formula".