How come the trivial solutions to the Riemann Hypothesis can be ignored, but a non-trivial solution would be a significant development?

The “trivial zeros” are the zeros produced using a simple algorithm. So, have we found some proof that there is no other algorithm that reliably produces zeros? If an algorithm were to be found which reliably produces zeros off the critical line, would these zeros simply be added to the set of trivial zeros and the search resumed as normal?