Bias and Randomness

After discussing how bias can enter the hiring process, someone brought up this wrinkle:

Suppose, that after analyzing your success at hiring, you realize that all your interviewing and selecting doesn't actually improve your selection of new hires. In other words, you realize (as the HR manager) that statistically, you discern that in predicting who will do well, who will stay with the company, you are doing no better than if you had randomly selected the individuals for hire (and your process is certainly introducing bias).

So, you propose, that instead of an expensive personal interviewing process, you conduct a few random phone interviews (to weed out those who can pass your written test but lack the ability to actually speak with another person), and then you place all remaining candidates into a pool, and randomly select those who will receive offers from your company. You argue that, since you can't actually discern which candidates will go on to successful careers at your company, and you are certainly (although inadvertently) introducing bias against some groups, randomly selecting the candidates for hire will remove bias, and do no worse in selecting potential employees.

Is your idea feasible? Is it recommendable (i.e., there won't be negative fallout that you can't address)? Will it be acceptable to the candidates themselves? To Society?