Just a glance at Rivera v Illinois  USSC No 07-9995 (31 March 2009) will get you thinking about whether there should be peremptory challenges to potential jurors.
There is huge variety in the jury selection procedures that jury trial jurisdictions have developed. Peremptory challenges are no longer allowed in England and Wales. Where such challenges are permitted, their number varies. In the USA there are measures to prevent peremptory challenges being exercised on racial or sexist grounds. That requires the judge to determine whether a peremptory challenge should be permitted.
In Rivera the judge wrongly refused to allow a peremptory challenge. The trial proceeded with that juror knowing she was not wanted by the accused. She became the foreman (foreperson) of the jury. The Supreme Court unanimously held that the error did not raise due process issues, as there was no risk of bias.
Perhaps the Court had to come to that conclusion, because to reverse the error here would have the effect of discouraging judges from ever exercising the control over the use of peremptory challenges that was designed to overcome racial or sexist discrimination in jury selection.
Be that as it may, Rivera illustrates the complications that can arise from sensitivity to the motives for which peremptory challenges may be exercised.
How big should juries be? How can it be ensured they are not biased? What sort of inquiry should be permitted into the competence of potential jurors? These issues go to the fundamentals of the jury trial process: the need for the facts at issue in a trial to be determined impartially.