I think that the discovery of a prime-generting algorithm would simply split the study up into a "list of largest known primes not generated by algorithm X" and "primes generated by algorithm X", much like the classification of finite simple groups.

Following this to its logical conclusion, even if we provably knew every algorithm generating $n$-digit primes in poly($n$)-time, we'd still be left with the question of which primes weren't generated by one of these functions, giving a concept of 'sporadic' prime, and it would then be these that were of interest.

I think that the discovery of a prime-generating algorithm would simply split the study up into a "list of largest known primes not generated by algorithm X" and "primes generated by algorithm X", much like the classification of finite simple groups.

Following this to its logical conclusion, even if we provably knew every algorithm generating $n$-digit primes in poly($n$)-time, we'd still be left with the question of which primes weren't generated by one of these functions, giving a concept of 'sporadic' prime, and it would then be these that were of interest.